THE   ROBERT   E.  COWAN  COLLECTION 

I'RKSKXTED    TO    THE 

UNIVERSITY  OF  CHLIFORNIfl 


C.  P.  HUNTINGTON 


dUNE,  1897. 


flccession  No./T?  /  3  ^       Class 


No, 


HEXOERSOX   &   HAMI.TN'S 


LIGHTNING  CALCULATOR; 


CONTAINING 


The  Shortest,  Siin/rfc-sf,  and  most  Raintf  Method  of  Computing  Numbers, 

to  uU  kinds  of  I»itsiness^  find  -wifhi.it  the  Coni/»re- 
heusion  of  every  one  htirinf/  the  sfif/htest 
knowledge  of  Figures. 


ENERGY  IS  THE  PRICE  OF  SUCCESS. 


"The  methods  of  calculation,  by  Prof.  J.  A.  HEXDF.USOX.  arc  invaluable  to  business  men,  and  will  prove  n 
light  u  all  coming  generations."— A.  -I.  W\i;M-:i;,  Pres.  Elmira  Commercial  Cell' 

••The  above  two  methods  are   the  finest   known  for   liyhtnin^  niultipftcatiOJl.1'-    I'rof.   1).   I!.  l-'olJD.  I-'emale 
.  Klmira. 

"I  have  examined  Prof.  J.  A.  HENDERSON'S  new  methods  of  calculation:  They  are  renuu-kable  for  originality, 
and  of  great  prtuttkul  value.  Ui.s  methods  of  calculating  Interest  are  peculiarly  dear  and  comprehensive  ill  their 
adaptation  to  all  possible  cases."— Rev.  DR.  O.  P.  FITZGERALD 


Address  all  orders  for   this  Book  to 

Prof.     J.     .A..     HENDERSON, 

SAN  FRANCISCO,  GAL. 


SAN    FRANCISCO: 

A.  L.  BANCROFT  &  CO.,  PRINTERS  AND  LITHOGRAPHERS, 

721  Market  Si; 

1872. 


i  ing  to  Act  of  '.-irian  at  Washington. 


HENDERSON  &  HAMLWS 


LIGHTNING  CALCULATOR; 


CONTAINING 


The  Shortest,  Simplest,  and  most  Rapid  Method  of  Computing  Numbers, 
adapted  to  all  kinds  of  Business,  and  within  the  Compre- 
hension of  every  one  having  the  slightest 
knowledge  of  Figures. 


ENERGY     IS     THE     PRICE     OF    SUCCESS, 


"The  methods  of  calculation,  by  Prof.  J.  A.  HENDERSON,  are  invaluable  to  business  men,  and  -will  prove  a 
light  in  science  to  all  coming  generations."— A.  J.  WARNER,  Pres.  Elmira  Commercial  College. 

"The  above  two  methods  are  the  finest  known  for  lightning  multiplication."— Prof.  D.  R.  FORD,  Female 
College,  Elmira. 

"I  have  examined  Prof.  J.  A.  HENDERSON'S  new  methods  of  calculation:  They  are  remarkable  for  originality, 
and  of  great  practical  value.  His  methods  of  calculating  Interest  are  peculiarly  clear  and  comprehensive  in  then- 
adaptation  to  all  possible  cases."— Rev.  DB.  O.  P.  FITZGERALD 


Address  all  orders  for  this  Book  to 

IProf.    J.     ^.    HENDERSON, 

SAN  FRANCISCO,  CAL. 


SAN    FRANCISCO: 

A.  L.  BANCKOFT  &  CO.,  PRINTERS  AND  LITHOGRAPHERS, 

721  Market  Street,  San  Francisco. 

1872. 


AGE 

7o  /3J~ 


It  is  better  to  know  everything  about  something,  than  some- 
thing about  everything. 

Early  ideas  are  not  usually  true  ideas,  but  need  to  be  revised 
and  re-revised.  Right  means  straight,  and  wrong  means  crooked. 
And  knowing  that  thought  kindles  at  the  fire  of  thought,  we  do 
not  hesitate  or  offer  any  apology  for  presenting  to  the  Public 
some  new  seed-thoughts,  and  right  methods  of  operation  in  busi- 
ness calculations.  The  practical  utility  of  this  book  is  found  in 
the  brevity  and  conciseness  of  its  rules.  Particular  attention  is 
invited  to  the  grand  improvements  in  the  subjects  of  computing 
time,  all  possible  cases  in  Interest,  Squaring  and  Multiplying 
Numbers,  Dividing  and  Multiplying  Fractions,  and  an  infinite  num- 
ber of  methods  of  Extracting  Square  and  Cube  Root. 


«A? 

THK 

JNIVERSITT 


ADDITION. 

To  be  able  to  add  two,  three,  or 
four  columns  of  figures  at  once  is 
deemed  by  many  to  be  a  Herculean 
task,  and  only  to  be  accomplished  by 
the  gifted  few;  or,  in  other  words,  by 
mathematical  prodigies.  If  we  can 
succeed  in  dispelling  this  illusion,  it 
will  more  than  repay  us  ;  and  we  feel 
very  confident  that  we  can,  if  the 
student  will  lay  aside  all  prejudice, 
bearing  steadily  in  mind  that  to  be- 
come proficient  in  any  new  branch  or 
principle,  a  little  wholesome  appli- 
cation is  necessary.  On  the  contrary, 
we  cannot  teach  a  student  who  takes 
no  interest  in  the  matter,  one  who 
will  always  be  a  drone  in  society. 
Such  men  have  no  need  of  this  prin- 
ciple. 

If  two,  three,  or  more  columns  can 
be  carried  up  at  a  time,  there  must 
be  some  law  or  rule  by  which  it  is 
done.  We  have  two  principles  of  Ad- 
dition ;  one  for  adding  short  columns, 
and  one  for  adding  very  long  columns. 
They  are  much  alike,  differing  only 
in  detail.  When  one  is  thoroughly 
learned,  it  is  very  easy  to  learn  the 
second.  By  a  little  attention  to  the 
following  example,  much  time  in  future 
will  be  saved, 

ADDITION   OF    SHORT    COLUMNS   OF   FIG- 
URES. 

Addition  is  the  basis  of  all  numerical 
operations,  and  is  used  in  all  depart- 
ments of  business.  To  aid  the  business 
man  in  acquiring  facility  and  accuracy 
in  adding  short  columns  of  figures,  the 
following  method  is  presented  as  the 
best  — 


PROCESS. — Commence  at 

274  the  right  hand  column,  add 

346  thus  :  16,  22,  32;  then  carry 

134  the  3  tens  to  the  second  column; 

342  then  add  thus:  7,  14,  25;  carry 

727  the  2  hundreds  to  the  third  col- 

329  umn,  and  add  the  same  way:  12, 

16,  21. 

2152. 

In  this  way  you  name  the  sum  of  two 
figures  at  once,  which  is  quite  as  easy 
as  it  is  to  add  one  figure  at  a  time,. 
Never  permit  yourself,  for  once,  to 
add  up  a  column  in  this  manner:  9 
and  7  are  16,  and  2  are  18  and  4  are 
22,  and  6  are  28,  and  4  are  32.  It 
is  just  as  easy  to  name  the  result  of 
two  figures  at  once,  and  four  times  as 
rapid. 

The  following  method  is  recom- 
mended for  the 

ADDITION    OF    LONG    COLUMNS    OF   FIGURES. 

In  the  addition  of  long  columns  of 
figures,  which  frequently  occur  in  books 
of  accounts,  in  order  to  add  them  with 
certainty,  and,  at  the  same  time,  with 
ease  and  expedition,  study  well  the  fol- 
lowing method,  which  practice  will  ren- 
der familiar,  easy,  rapid,  and  certain. 

THE    EASY  WAY    TO   ADD. 
EXAMPLE    2 EXPLANATION. 

Commence  at  9  to  add,  and  add  as 
near  20  as  possible,  thus  :  9+2+4+ 
3=18,    place    the   8   to   the   right    of 
the  3,  as  in  example  ;  commence     T 
at  6  to  add  6+4+8=18;    place     4 
the  8  to  the  right  of  the  8,  as  in     6 
example  ;  commence  at  6  to  add     36 
6-f-4-j-7=17  ;  place  the  7  to  the     9 
right  of    the   7,   as    in   example;     4 
commence   at  4   to   add  4+9+3     77 
=16  ;  place  the  6  to  the  right  of    4 
the  3,  as  in  example  ;  commence     6 
at  6  to  add   6+4+7=17  ;  place    88 


the  7  to  the  right  of  the  7  as  in    4 
example;  now,  having  arrived  at    6 
the  top  of  the  column,   we    add     38 
and  figures  in  the  new   column, 
thus:    7-f6-{-7+8-f-8=36;    place    4 
the    right-hand     figure     of    36,     2 
which  is  a  6,  under  the  original     9 
column,  as  in  example,   and  add  — 
the   left-hand  figure,  which  is    a  86 
3,  to  the  number  of  figures  in  the 
new  column;    there  are    5  figures    in 
the  new   column,   therefore    3+5=8; 
prefix   the   8   with  the   6,   under    the 
original   column,  as  in  example;   this 
makes   86,    which  is  the   sum   of    the 
column. 

Remark  1. — If ,  upon  arriving  at  the 
top  of  the  column,  there  should  be 
one,  two  or  three  figures  whose  sum 
will  not  equal  10,  add  them  on  to  the 
sum  of  the  figures  of  the  new  column, 
never  placing  an  extra  figure  in  the 
new  column,  unless  it  be  an  excess  of 
units  over  ten. 

Remark  2.  —  By  this  system  of 
addition  you  can  stop  at  any  place  in 
the  column,  where  the  sum  of  the 
figures  will  equal  10  or  the  excess  of 
10;  but  the  addition  will  be  more 
rapid  by  your  adding  as  near  20  as 
possible,  because  you  will  save  the 
forming  of  extra  figures  in  your  new 
column. 

EXAMPLE — EXPLANATION. 

2+6+7=15,   drop   10,  place  the   5 
to  the   right   of    the   7;    6+5+4=15, 
drop   10,  place  the  5  to   the  right   of 
the   4,    as    in   example;    8+3+7=18, 
drop  10,  place  the   8  to  the  right     4 
of  the  7,  as   in  example;  now  we     78 
have  an   extra  figure,  which  is  4;     3 
add  this  4  to  the  top  figure  of  the     8 
new  column,  and  this  sum  on  the     4s 
balance  of  the  figures  in  the  new     5 


column,     thus:     4+8+5+5=22;     6 
olace  the  right-hand  figure  of  22     7s 
under  the  original  column,  as  in     6 
example,    and  add  the  left-hand     2 
Eigure  of  22  to  the  number  of  fig — 
ures  in   the  new  column,    which  52 
are    three,   thus:    2+3=5;  prefix 
this  5  to  the  figure  2,  under  the  orig- 
inal column;  this  makes  52,  which  is 
the  sum  of  the  column, 

RULE. — For  'adding  two  or  more  col- 
umns, commence  at  the  right-hand,  or 
units'  column;  proceed  in  the  same  man- 
ner as  in  adding  one  column;  after 
the  sum  of  the  first  column  is  obtained, 
add  all  except  the  right-hand  figure 
of  this  sum  to  the  second  column, 
adding  the  second  column  the  same 
way  you  added  the  first;  proceed  in 
like  manner  with  all  the  columns, 
always  adding  to  each  successive 
column  the  sum  of  the  column  in  the 
next  lower  order,  minus  the  right-hand 
figure, 

N.  B.  The  small  figures  which  we 
place  to  the  right  of  the  column  when 
adding  are  called  intergers 

The  addition  by  intergers,  or  by 
forming  a  new  column,  as  explained 
in  the  preceding  examples,  should  be 
used  only  in  adding  very  long  columns 
of  figures,  say  a  long  ledger  column, 
where  the  footings  of  each  column 
would  be  two  or  three  hundred,  in 
which  case  it  is  superior  and  much 
more  easy  than  any  other  mode  of 
addition;  but  in  adding  short  columns 
it  would  be  useless  to  form  an  extra 
column,  where  there  is  only,  say  six  or 
eight  figures  to  be  added.  In  making 
short  additions,  the  following  sugges- 
tions will,  we  trust,  be  of  use  to  the 
accountant  who  seeks  for  information 
on  this  subject. 


IVERSITY 


In  the  addition  of  several  columns 
of  figures,  where  there  are  only  four  or 
five  deep,  or  when  their  respective 
sums  will  range  from  twenty-five  to 
forty,  the  accountant  should  com- 
mence with  the  unit  column,  adding 
the  sum  of  the  first  two  figures  to  the 
sum  of  the  next  two,  and  so  on, 
naming  only  the  results,  that  is,  the 
sum  of  every  two  figures, 

In  the .  present  example,  in  346 
adding  the  unit  column  instead  235 
of  saying  8  and  4  are  12  and  5  724 
are  17  and  6  are  23,  it  is  better  598 
to  let  the  eye  glide  up  the  col- 
umn, reading  only,  8,  12,  17,  23; 
and  still  better,  instead  of  making .  a, 
separate  addition  for  each  figure, 
group  the  figures  thus:  12  and  11 
are  23,  and  proceed  in  like  manner 
with  each  column.  For  short  columns 
this  is  a  very  expeditious  way,  and 
indeed  to  be  preferred,  but  for  long 
columns,  the  addition  by  integers  is 
the  most  useful,  as  the  mind  is  relieved 
at  intervals,  and  the  mental  labor  of 
retaining  the  whole  amount,  as  you 
add,  is  avoided,  which  is  very  impor- 
tant to  any  person  whose  mind  is  con- 
stantly -employed  in  various  commer- 
cial calculations 


In  adding  a  long  column,  where  the 
figures  are  of  a  medium  size,  that  is, 
as  many  8s  and  9s  as  there  are  2s 
and  Hs,  it  is  better  to  add  about  three 
figures  at  a  time,  because  the  eye  will 
distinctly  see  that  many  at  once,  and 
the  ingenious  student  will  in  a 
short  time,  if  he  adds  by  integers,  be 
able  to  read  bhe  amount  of  three  fig- 
ures at  a  glance,  or  as  quick,  we 
mi(^ht  say,  as  he  would  read  a  single 
figure. 


Here  we   begin    to   add   at  the  5268 
bottom   of   the   unit   column  and     67 
add     successively     three     figures    43 
at    a    time,   and    place  their    re-     384 
spective   sums,  minus   10,   to  the   e54 
right    of    the   last  figure  added  ;     62 
if    the     three     figures     do      not     87* 
make   10,    add   on    more   figures;   565 
if  the  three   figures   make   20   or    53 
more,    only   add   two   of  the  fig-    444 
ures.      The     little     figures     that    877 
are  placed  to  the   right  and  left    33 
of  the   column    are    called    inte-     844 
gers.       The      integers      in      the    356 
present     example,    belonging    to     14 
the   units'   column,    are   4,    4,  5,  - 
4,    6,    which    we     add     together    803 
making  23;  place  down  3  and  add 
2    to   the   number   of  integers,   which 
gives  7,  which  we  add  to  the  tens  and 
proceed  as  before. 


KEASON. — In  the  above  example, 
every  time  we  placed  down  an  in- 
teger we  discarded  a  ten,  and  when 
we  set  down  the  3  in  the  answer  we 
discarded  two  tens;  hence,  we  add  2 
on  to  the  number  of  integers  to  ascer- 
tain how  many  tens  were  discarded; 
there  being  5  integers,  it  made  7  tens, 
which  we  now  add  to  the  column  of 
tens;  on  the  same  principle  we  might 
add  between  20  and  30,  always  set- 
ting down  a  figure  before  we  got  to 
30;  then  every  integer  set  down 
would  count  for  2  tens,  being  dis- 
carded in  the  same  way,  it  does  in 
the  present  instance  for  one  ten. 
When  we  add  between  10  and  20,  and 
in  very  long  columns,  it  would 
be  much  better  to  go  as  near  30  as 
possible,  and  count  2  tens  for  every 
integer  set  down,  in  wnich  case  we 
would  set  down  about  one-half  as 


many  integers  as  when  we  write  an  in- 
teger for  every  ten  we  discard 

"When  adding  long  columns  in  a 
ledger  or  day-book,  and  where  the  ac- 
countant wishes  to  avoid  the  writing  of 
extra  figures  in  the  book,  he  can  place 
a  strip  of  paper  alongside  of  the  col- 
umn he  wishes  to  add,  and  write  the 
integers  on  the  paper,  and  in  this  way 
the  column  can  be  added  as  conven- 
iently almost  as  if  the  integers  were 
written  in  the  book. 

Perhaps,  too,  this  would  be  as 
proper  a  time  as  any  other  to  urge 
the  importance  of  another  good  habit; 
I  mean  that  of  making  plain  figures. 
Some  persons  accustom  themselves  to 
making  mere  scrawls,  and  impor- 
tant blunders  are  often  the  result.  If 
letters  be  badly  made,  you  may  judge 
from  such  as  are  known;  but  if 
one  figure  be  illegible,  its  value  can- 
not be  inferred  from  the  others.  The 
vexation  of  the  man  who  wrote  for  2 
or  3  monkeys,  and  had  203 ^ent  him, 
was  of  far  less  importance  than 
errors  and  disappointments  some- 
times resulting  from  this  inexcusable 
practice. 

"We  will  now  proceed  to  give  some 
methods  of  proof.  Many  persons  are 
fond  of  proving  the  correctness  of 
work,  and  pupils  are  often  instructed 
to  do  so,  for  the  double  purpose  of 
giving  them  exercise  in  calculation  and 
saving  their  teacher  the  trouble  of  re- 
viewing their  work. 

There  are  special  modes  of  proof 
of  elementary  operations,  as  by  cast- 
ing out  threes  or  nines,  or  by  chang- 
ing the  order  of  the  operation,  as  in 
adding  upward  and  then  downward. 
In  addition,  some  prefer  reviewing 
the  work  by  performing  the  Addition 


downward,  rather  than  repeating  the 
ordinary  operation.  This  is  better, 
for  if  a  mistake  be  inadvertently 
made  in  any  calculation,  and  the  same 
routine  be  again  followed,  we  are 
very  liable  to  fall  again  into  the  same 
error.  If,  for  instance,  in  running 
up  a  column  of  Addition  you  should 
say  84  and  8  are  93,  you  would  be 
liable,  in  going  over  the  same  again, 
in  the  same  way  to  slide  insensibly 
into  a  similar  error;  but  by  begin- 
ning at  a  different  point  this  is 
avoided. 

This  fact  is  one  of  the  strongest 
objections  to  the  plan  of  cutting  off 
the  upper  line  and  adding  it  to  the 
sum  of  the  rest,  and  hence  some  cut 
off  the  lower  line  by  which  the  spell 
is  broken.  The  most  thoughtless  can- 
not fail  to  see  that  adding  a  line  to 
the  sum  of  the  rest  is  the  same  as  add- 
ing it  in  with  the  rest. 

The  mode  of  proof  by  casting  out 
the  nines  and  threes  will  be  fully  ex- 
plained in  a  following  chapter. 

A  very  excellent  mode  of  avoiding 
error  in  adding  long  columns  is  to 
set  down  the  result  of  each  column 
on  some  waste  spot,  observing  to 
place  the  numbers  successively  a 
place  further  to  the  left  each  time, 
as  in  putting  down  the  product  fig- 
ures in  multiplication  ;  and  afterward 
add  up  the  amount.  In  this  way  if 
the  operator  lose  his  count,  he  is  not 
compelled  to  go  back  to  units,  but 
only  to  the  foot  of  the  column  011 
which  he  is  operating.  It  is  also 
true  that  the  brisk  accountant,  who 
thinks  on  what  he  is  doing,  is  less 
liable  to  err  than  the  dilatory  one, 
who  allows  his  mind  to  wander. 
Practice,  too,  will  enable  a  person  to 


read    accounts    without  naming  each 
figure :  thus,  instead  of  saying  8  and  6 
are  14,  and  7  are  21  and  5  are  26,  it  is 
better  to  let  the  eye   glide   up  the  col- 
umn,   reading    only    8,    14,    21,    26, 
etc. ;  and,  still  further,  it  is  quite 
practicable  to  accustom  one's  self  87 
to   group  the  figures  in  adding,  23 
and   thus   proceed   very   rapidly.   45 
Thus  in  adding  the  units'  column,  62 
instead   of  adding   a  figure   at   a  24 
time,  we   see  at   a   glance  that  4  - 
and  2  are  6,  and  that  5  and  3  are 
8;  then  6  and  8   are  14;  we   may  then, 
if  expert,  add   constantly  the  sum  of 
two  or  three   figures   at   a   time,  and 
with  practice  this  will  be  found  highly 
advantageous  in  long  columns   of  fig- 
ures;   or  two    or   three   columns   may 
be    added   at   a   time,    as     the     prac- 
tised eye  will  see  that   24  and   62  are 
86   almost   as   readily  as  that  4  and  2 
are  6, 


MULTIPLICATION. 

Multiplication,  in  its  most  general 
sense,  is  a  series  of  additions  of 
the  same  number;  therefore,  in  mul- 
tiplication, a  number  is  repeated  a 
certain  number  of  times,  and  the  re- 
sult thus  obtained  is  called  the  prod- 
uct. When  the  multiplicand  and  the 
multiplier  are  each  composed  of  only 
two  figures,  to  ascertain  the  product, 
we  have  the  following 

RULE.  —  Set  down  the  smaller  fac~ 
tor  under  the  larger,  units  under 
units,  tens  under  tens.  Begin  with 
the  unit  figure  of  the  multiplier,  mul- 
tiply by  it,  first  the  units  of  the 
multiplicand,  setting  the  units  of  the 
product,  and  reserving  the  tens  to  ie 
added  to  the  next  product;  now  mul- 
tiply the  tens  of  the  multiplicand  by 
the  unit  figure  of  the  multiplier,  and 
the  units  of  the  multiplicand  by  tens 


figure  of  the  multiplier;  add 
two  products  together,  setting  down 
the  units  of  their  sum,  and  reserving 
the  tens  to  be  'added  to  the  next  prod- 
uct; now  multiply  the  tens  of  the 
multiplicand  by  the  tens'  figure  of  the 
multiplier,  and  set  down  the  whole 
amount.  This  will  be  the  complete 
product. 

Remark. — Always  add  in  the  tens 
that  are  reserved  as  soon  as  you  form 
the  first  product. 

EXAMPLE    1. EXPLANATION. 

1.  Multiply  the  units  of  the  24 
multiplicand  by  the  unit  fig-  31 

ure   of    the    multiplier,    thus:   1     

X4  is  4;  set  the  4  down  as  in  744 
example.  2.  Multiply  the  tens 
in  the  multiplicand  by  the  unit  figure 
in  the  multiplier,  and  the  units  in  the 
multiplicand  by  the  tens  figure  in  the 
multiplier,  thus:  1x2  is  2;  3x4 
are  12,  add  these  two  products  to- 
gether, 2  plus  12  are  14,  set  the  4 
down  as  in  example,  and  reserve  the 
1  to  be  added  to  the  next  product. 
3.  Multiply  the  tens  in  the  multipli- 
cand by  the  tens'  figures  in  the  mul- 
tiplier, and  add  in  the  tens  that  were 
reserved,  thus;  3x2  are  6,  and  6  plus 
1  equal  7;  now  set  down  the  whole 
amount,  which  is  7. 

EXAMPLE    1. EXPLANATION. 

Multiply  first  upper  by  units,  123 
5x3  are  15,  set  down  the  5,  re-  45 

serve  the  1  to  carry  to  the  next  

product;  now  multiply  second  5535 
upper  by  units  and  first  upper  by  tens, 
5X2  are  10,  plus  1  are  11,  4x3  are 
12,  add  these  products  together;  11 
plus  12  are  23,  set  clown  the  3,  re- 
serve the  2  to  carry;  now  multiply 
third  upper  by  units,  and  second  up- 
per by  tens,  add  these  two  products 
together,  always  adding  on  the  re- 


8 


served  figure  to  the  first  product;  5 
Xl  are  5,  plus  2  are  7,  4X2  are  8,  and 
7  plus  8  are  15,  set  down  the  5,  re- 
serve the  1 ;  now  multiply  third  upper 
by  tens,  and  set  down  the  whole 
amount;  4x1  are  4  plus  1  are  5,  set 
down  the  5.  This  will  give  the  com- 
plete product. 

Multiply  32  by  45  in  a  single  line. 
Here  we  multiply  5X2  and  set  32 
down  and  carry  as  usual;  tljen  to  45 
what  you  carry  add  5X3  and  4X 


2,  which   gives   24;    set  down  4  1440 
and  carry  2  to  4X3,  which  gives 

14  and  completes  the  product. 

Multiply  123  by  456  in  a  single 
line. 

Here  the  first  and  second  123 
places  are  found  as  before;  for  456 
the  third,  add  6X1,  5x2,  4X  - 

3,  with  the  2  you  had  to  carry,  56088 
making    30;     e?t    down   0    and 
carry   3;    then   drop   the   units3    place 
and  multiply   the  hundreds  and  tens 
crosswise,     as    you   did  the   tens   and 
units,  and  you  find   the   thousand   fig- 
ure;   then,    dropping  both  units   and 
tens,  multiply  the  4X1,  adding   the  1 
you  carried,   and  you  have   5,    which 
completes    the     product.      The    same 
principle  may  be  extended  to  any  num- 
ber  of  places;    but  let   each   step  be 
made  perfectly  familiar  before  advanc- 
ing to  another.  Begin  with  two  places, 
then  take  three,  then   four,  but  always 
practising  some  time  on  each    number, 
for  any  hesitation  as  you   progress  will 
confuse  you. 

CURIOUS  AND   USEFUL   CONTRACTIONS. 

To  multiply  any  number,  of  two  fig- 
ures, by  11. 

RULE. — Write  the  sum  of  the  figures  be- 
tween them. 

I.     Multiply  45  by  11.     Ans.  495. 


Here  4  and  5  are  9,  which  write  be* 
tween  4  and  5, 

2.  Multiply  34  by  11.     Ans.  374.     , 
N.  B.     When  the   sum   of  the  two 

figures  is  over  9,  increase  the  left-hand 
figure  by  the  1  to  carry, 

3.  Multiply  87  by  11.     Ans.  957. 

To  square  any  number  of  9s  in- 
stantaneously, and  without  multiply- 
ing. 

RULE. — Write  down  as  many  9s 
less  one  as  there  are  9s  in  the  given 
number,  an  8,  as  many  Os  as  9s, 
and  a  1. 

4.  What  is   the    square    of    9999? 
Ans.  99980001. 

EXPLANATION. — We  have  four  9s  in 
the  given  number,  so  we  write  down 
three  9s,  then  an  8,  then  three  Os,  and 
al. 

5.  Square  999999.     Answer  999998- 
000001. 

To  square  any   number   ending  in  5. 

RULE. — Omit  the  5  and  multiply 
the  number  as  it  will  then  stand  by 
the  next  higher  number,  and  annex  25 
to  the  product, 

6.  What  is  the  square  of  75?     Ans. 
5625. 

EXPLANATION.  —  We  simply  say,  7 
times  8  are  56,  to  which  we  annex 
25. 

7.  What  is  the  square  of  95?    Ans. 
9025. 

PRACTICAL   BUSINESS     METHOD 

For  Multiplying  all  Mixed  Numbers. 

Merchants,  grocers,  and  business 
men  generally,  in  multiplying  the 
mixed  numbers  that  arise  in  the  prac- 
tical calculations  of  their  business, 
only  care  about  having  the  answer 
correct  to  the  nearest  cent;  that  is, 
they  disregard  the  fraction.  WThen 
it  is  a  half  cent  or  more,  they  call  it 


9 


another  cent,  if  less  than  half  a  cent, 
they  drop  it.  And  the  object  of  the 
following  rule  is  to  show  the  business 
man  the  easiest  and  most  rapid  pro- 
cess of  finding  the  product  to  the  nearest 
unit  of  any  two  numbers,  one  or  both 
of  which  involves  a  fraction. 

GENEBAL  RULE. 

To  multiply  any  two  numbers  to  the 
nearest  unit. 

1st.  Multiply  the  whole  number  in 
the  multiplicand  by  the  fraction  in  the 
multiplier  to  the  nearest  unit, 

2d.  Multiply  the  whole  nmmber  in  the 
multiplier  by  the  fraction  in  the  multipli- 
cand to  the  nearest  unit. 

3d.  Multiply  the  whole  numbers  to- 
gether and  add  the  three  products  in  your 
mind  as  you  proceed. 

N.  B.  In  actual  business  the  work 
can  generally  be  done  mentally,  for 
only  easy  fractions  occur  in  business. 

N.  B.  This  rule  is  so  simple  and  so 
true,  according  to  all  business  usage, 
that  every  accountant  should  make  him- 
self perfectly  familiar  with  its  ap- 
plication. There  being  no  such  thing 
as  a  fraction  to  add  in,  there  is 
scarcely  any  liability  to  error  or  mis- 
take. By  no  other  arithmetical  pro- 
cess can  the  result  be  obtained  by  so 
few  figures. 

EXAMPLE  FOR  MENTAL   OPERATION. 

Multiply  11J  by  8J  by  business 
method. 

Here  J  of  11  to   the   nearest 
unit  is  3,   and  J    of    8    to    the     11 J 
nearest  unit  is  3,  making  6,  so      8J 

we   simply  say,   8  times   11   are  

88  and  6  are  94.  Ans.  94 

REASON. — \  of  11  is  nearer  3  than 
2,  and  J  of  8  is  nearer  3  than  2, 
Make  the  nearest  whole  number  the 
quotient. 


A  VALUABLE  HINT  TO  MERCHANTS  AND  ALL 
RETAIL  DEALERS  IN  FOREIGN  AND  DOMES- 
TIC DRY  GOODS. 

Retail  merchants,  in  buying  goods 
by  wholesale,  buy  a  great  many  ar- 
ticles by  the  dozen,  such  as  boots 
and  shoes,  hats  and  caps,  and  notions 
of  various  kinds.  Now,  the  mer- 
chant, in  buying,  for  instance,  a 
dozen  hats,  knows  exactly  what  one  of 
those  hats  will  retail  for  in  the  mar- 
ket where  he  deals ;  and,  unless 
he  is  a  good  accountant,  it  will  often 
take  him  some  time  to  determine 
whether  he  can  afford  to  purchase  the 
dozen  hats  and  make  a  living  profit 
in  selling  them  by  the  single  hat; 
and  in  buying  his  goods  by  auction, 
as  the  merchant  often  does,  he  has 
not  time  to  make  the  calculation  be- 
fore the  goods  are  cried  off.  He, 
therefore,  loses  the  chance  of  making 
good  bargains  by  being  afraid  to  bid 
at  random,  or  if  he  bids,  and  the 
goods  are  cried  off,  he  may  have 
made  a  poor  bargain  by  bidding  thus 
at  a  venture.  It  then  becomes  a 
useful  and  practical  problem  to  de- 
termine instantly  what  per  cent,  he 
would  gain  if  he  retailed  the  hats  at  a 
certain  price. 

RAPID   PROCESS   OF   MARKING   GOODS 

To  tell  what  an  article  should  retail 
for  to  make  a  profit  of  20  per  cent,  is 
done  by  removing  the  decimal  point 
one  place  to  the  left. 

For  instance,  if  hats  costs  $17.50 
per  dozen,  remove  the  decimal  point 
one  place  to  the  left,  making  $1.75, 
what  they  should  be  sold  for  a  piece 
to  gain  20  per  cent,  on  the  cost.  If 
they  cost  $31.00  per  dozen,  they 
should  be  sold  for  $3.10  apiece,  etc. 


10 


We  take  20  per  cent  as  the  basis,  for 
the  following  reasons,  namely:  be- 
cause we  can  determine  instantly,  by 
simply  removing  the  decimal  point, 
without  changing  a  figure;  and,  if 
the  goods  would  not  bring  at  least  20 
per  cent,  profit  in  the  home  market, 
the  merchant  could  not  afford  to  pur- 
chase and  would  look  for  goods  at 
lower  figures. 

Now,  as  removing  the  decimal  point 
one  place  to  the  left,  on  the  cost  of  a 
dozen  articles  gives  the  selling  price 
of  a  single  one  with  20  per  cent, 
added  to  the  cost,  and,  as  tile  cost  of 
any  article  is  100  per  cent. ,  it  is  ob- 
vious that  the  selling  price  would  be 
20  per  cent,  more,  or  120  per  cent; 
hence,  to  find  50  per  cent,  profit, 
which  would  make  the  selling  price 
150  per  cent.,  we  would  first  find 
120  per  cent.,  then  add  30  per  cent., 
by  increasing  it  one-fourth  itself;  to 
make  40  per  cent.,  add  20 per  cent.,  by 
increasing  it  ore-sixth  itself;  for  35 
per  cent,  increase  it  one-eighth  itself, 
etc.  Hence,  to  mark  an  article  at  any 
per  cent,  profit,  we  have  the  follow- 
ing 

GENERAL    RULE. 

First  find  20  per  cent,  profit  by 
removing  the  decimal  point  one  place  to 
the  left  on  the  price  the  articles  cost  a 
dozen;  then,  as  20  per  cent,  profit  is  120 
percent.,  add  to,  or  substr act  from,  this 
amount  the  fractional  part  that  the  re- 
quired per  cent,  added  to  100  is  more  or 
less  than  120. 

TABLE. 

For  Marking  all  Articles  bought  by  the 
Dozen. 

N.  B.    Most  of  these  are  used  in  business. 

To  make  20  pr  ct.  remove  the  point  one  place  to  the  left. 


To  make  33  %  p  ct.  remove  point  and  add  one-ninth  itself. 

one-tenth       '• 

30 
28 
26 
•25 


one- twelfth    " 
one-fifteenth 
one-twentieth ' 
one-twerx"-fourth 
subtr  ct  one-sixteenth 
one-thirty-sixth 
one-ninety-sixth 


80 

60 

50 

44 

40 

37% 

35 


and  add  one-half  itself, 
one-third 
one-fourth 
one-fifth 
one-sixth 
one-seventh 
one-eighth 


18  ?i 

If  I  buy  1  doz.  shirts  for  $28.00, 
what  shall  I  retail  them  for  to  make 
50  per  ct.V  Ans.  $3.50. 

EXPLANATION.  — Remove  the  point  one 
place  to  the  left,  and  add  on  J  it- 
self. 

Where  the  Multiplier  is  an  Aliquot  part 
of  100. 

Merchants  in  selling  goods  gener- 
ally make  the  price  of  an  article  some 
aliquot  part  of  100,  as  in  selling 
sugar  at  12 J  cents  a  pound  or  8 
pounds  for  1  dollar,  or  in  selling 
calico,  for  16  2-3  cents  a  yard  or  6 
yards  for  1  dollar,  etc.  And  to  be- 
come familiar  with  all  the  aliquot  parts 
of  100,  so  that  you  can  apply  them 
readily  when  occasion  requires,  is 
perhaps  the  most  useful,  and,  at  the 
same  time,  one  of  the  easiest  arrived  at 
of  all  the  computations  the  ac- 
countant must  perform  in  the  prac- 
tical calculations  of  the  counting- 
room. 

TABLE. 

Of  the  Aliquot  parts  of  100  and 
1000. 

N.  B.    Most  of  these  are  used  in  business. 

12%  is  J$  part  of  100.  8>j  is  1-12  part  of  100 

25      is  2-8  or  %  of  100.  16 8$  is  2-12  or  1-6  of  100 

37%  is  3-8  part  of  100.  33^  is  4-12  or  %  of  100 

50     is  4-8  or  %  of  100.  6G2s  is  8-12  or  %  of  100 

62%  is  %  part  of  100.  88^  is  10-12  or  5-6  of  100 

75  is  6-8  or  3£  of  100.  125  is  X  part  of  1000 

87%  is  %  part  of  100.  250  is  2-8  or  %  of  1000 

6%  is  1-16  part  of  100.  375  is   %  part  of  1000 

1834  is  3-16  part  of  100.  625  is  5/8  part  of  1000 

31%  is  5-16  part  of  100.  875  is  %  part  of  1000 

To  multiply  by  an  aliquot  part  of 
100. 

RULE. — Add  two  ciphers  to  the  multi- 
plicand, then  take  such  part  of  it  as  the 
multipliers  is  part  of  100. 

N.  B.  If  the  multiplicand  is  a 
mixed  number  reduce  the  fraction  to 


11 


"Cl  B  f. 

OP  TJTK 

UNIVERSITY 


a  decimal  of  two  places  before  divid- 
ing. 

General  Rules  for  Cancellation. 

RULE  IST.  Draw  a  perpendicular 
line  ;  observe  this  line  represents  the 
sign  of  equality.  On  the  right-hand 
side  of  this  line  place  dividends  only  ; 
on  the  left  hand  side  place  divisors 
only;  having  placed  dividends  on 
the  right  and  divisors  on  the  left  as 
above  directed, 

2d.  Notice  whether  there  are 
ciphers  both  on  the  right  and  left  of 
the  line  ;  if  so,  erase  an  equal  number 
from  each  side. 

3d.  Notice  whether  the  same  num- 
ber stands  both  on  the  right  and  left 
of  the  line  ;  if  so,  erase  them  both. 

4th.  Notice  again  if  any  number 
on  either  side  of  the  line  will  divide 
any  number  on  the  opposite  side 
without  a  remainder  ;  if  so,  divide 
and  erase  the  two  numbers,  retaining 
the  quotient  figure  on  the  side  of  the 
larger  number. 

5th.  See  if  any  two  numbers,  one  on 
flach  side,  can  be  divided  by  any  as- 
sumed number  without  a  remainder  ; 
if  so,  divide  them  by  that  number, 
and  retain  only  their  quotients. 
Proceed  in  the  same  manner  as  far  as 
practicable,  then, 

6th.  Multiply  all  the  numbers  re- 
maining on  the  right-hand  side  of  the 
line  for  a  dividend,  and  those  remain- 
ing on  the  left  for  a  divisor. 

7th.  Divide,  and  the  quotient  is  the 
answer. 

SIMPLE   INTEREST   BY   CANCELLATION. 

RULE.  —  Place  the  principal,  time 
and  rate  per  cent,  on  the  right-hand 
side  of  the  line.  If  the  time  consists 
of  years  and  months,  reduce  them  to 
months,  and  place  12  (the  number  of 
months  in  a  year)  on  the  left-hand 


side  of  the  line.  Should 
sist  of  months  and  days,  reduce  them 
to  days  or  decimal  parts  of  a  month. 
If  reduced  to  days,  place  36  on  the 
left.  If  to  decimal  parts  of  a  month, 
place  12  only,  as  before. 

Point  off  two  decimal  places  when 
the  time  is  in  months,  and  three  decimal 
places  when  the  time  is  in  days. 

NOTE.  —  If  the  principal  contains 
cents,  point  off  four  decimal  places 
when  the  time  is  in  months,  and  five 
decimal  places  when  the  time  is  in 
days, 

NOTE.  —  We  place  36  on  the  left 
because  there  are  360  interest  days  in 
a  year.  (Custom  has  made  this  law- 
ful.) 

LIGHTNING   METHOD 

OF 
COMPUTING    INTEREST. 

On  all  notes  that  bear  $12  per  an- 
num, or  any  aliquot  part  or  multiple 
of  $12. 

If  a  note  bears  $12  per  annum,  it 
will  certainly  ^"bear  $1  per  month  : 
hence  the  time  in  months  would  be 
the  interest  in  $  ;  and  the  decimal 
parts  of  a  month  would  be  the  in- 
terest in  decimal  parts  of  a  $  ;  there- 
fore when  the  note  bears  $12  per 
annum  we  have  the  following  rule  : 

RULE.  —  Reduce  the  years  to  months, 
add  in  the  given  months,  and  place 
one-third  of  the  days  to  the  right  of 
this  number,  and  you  have  the  interest  in 
dimes. 

EXAMPLE  1.  —  Required  the  interest 
of  $200  for  3  years,  7  months,  and  12 
days,  at  6  per  cent. 

200  y*  of  12  days  =  4. 

6 

--  Yr.  Mo.  Da. 

$12.00  ==  int.  for  1  yr.      37      12  =  43.  4  mo. 
Hence  43.4  dimes,  or  $43.40cts.,  Ans 

We  see  by  inspection  that  this 
note  bears  $12  interest  a  year;  hence 


12 


the  time  reduced  to  months,  with 
one-third  of  the  days  to  the  right,  is 
the  interest  in  dimes.  If  this  note 
bore  $6  a  year,  instead  of  $12,  we 
would  take  one-half  of  the  above  in- 
terest; if  it  bore  $18  instead  of  $12, 
we  would  add  one-half;  if  it  bore 
$24,  instead  of  $12,  we  would  multiply 
by  2,  etc. 

EXAMPLE  2.  —  Required  the  interest 
of  $150  for  two  years,  5  months,  and  13 
days,  at  8  per  cent. 

%  of  13  days  =4% 


150 
8 


Yr.  Mo.  Da. 


$12.00  =  int.  for  1  yr.    2    5    13  =  29.  4y3mos. 
Hence  $29.  4%  dimes,  or  $29.433%cts.,  Ans. 

We  see  by  inspection  that  this 
note  bears  $12  interest  a  year;  hence 
the  time  reduced  to  months,  with 
one-third  of  the  days  placed  to  the 
right,  gives  the  interest  at  once. 

EXAMPLE  3.  —  Required  the  interest 
of  $160  for  11  years,  11  months,  and  11 
days,  at  7j-  per  cent. 

160  %of  11  days  =  3%- 

7% 

Tr.  Mo,  Da. 

$12.  00  =  int.  for  1  yr.     11     11  11  =  143%mos. 
Hence  143.3%  dimes,  or  $143.36%cts.,  Ans. 

When  the  interest  is  more  or  less 
than  $12  a  Tear. 

RULE.  —  First  find  the  interest  for 
the  given  time  on  the  base  of  $12  in- 
terest a  year;  then,  if  the  interest  on  the 
note  is  only  $6  a  year,  divide  by  2;  if 
$24  a  year,  multiply  by  2;  if  $18  a  year, 
add  on  one-half,  etc. 

EXAMPLE  1.  —  "What  is  the  interest 
of  $300  for  4  years,  7  months,  and  18 
days,  at  6  per  cent.? 

%  of  18  days  =6. 

300  4yr.  7mo.  18da.=55.6mo. 

6 

$18.00  =int.  for  1  yr.    2)55.6,  int.  -at  $12  a  yr. 
$18=1%  times  $12.  27.8. 

$83.4.  Ans. 


If  the  interest  was  $12  a  year, 
$55.60  would  be  the  answer;  because 
55.6  is  the  time  reduced  to  months; 
but  it  bears  $18  a  year,  or  1J  times 
12;  hence  1J  times  55.6  gives  the 
interest  at  once. 

EXAMPLE  2. — Required  the  interest 
of  $150  for  three  years,  9  months,  and 
27  days,  at  four  per  cent 


150 
4 

$6.00  =  int.  for  1  yr. 
$6  =  %  times  S12. 


%  of  27  days  =9. 
3yr.  9mo.  27da  =  45.9mo. 
2)45.9.  int.  at  $12  a  year. 

$22.95,  Ans. 


If  the  interest  was  $12  a  year, 
$45.90  would  be  the  answer;  because 
$45.9  is  the  time  reduced  to  months  ; 
but  it  bears  $6  a  year,  or  \  times  12; 
hence  \  times  45.9  gives  the  interest  at 
once. 


RULES    FOE    DETERMINING    THE   WEIGHT    OF 
LIVE     CATTLE. 

Measure  in  inches  the  girth  round 
the  breast.,  just  behind  the  shoulder- 
blade,  and  the  length  of  the  back  from 
the  tail  to  the  fore  part  of  the  shoulder- 
blade.  Multiply  the  girth  by  the 
length,  and  divide  by  144.  If  the 
girth  is  less  than  three  feet,  multiply 
the  quotient  by  11 ;  if  between  three 
feet  and  five  feet,  multiply  by  16;  if 
between  five  feet  and  seven  feet, 
multiply  by  23',  if  between  seven 
and  nine  feet,  multiply  by  31.  If 
the  animal  is  lean,  deduct  l-20th 
from  the  result. 

Take  the  girth  and  length  in  feet, 
multiply  the  square  of  the  girth  by 
the  length,  and  multiply  the  product 
by  3.36.  The  result  will  be  the  an- 
swer in  pounds.  The  live  weight, 
multiplied  by  605,  gives  a  near  ap- 
proximation to  the  net  weight. 


13 


ASTRONOMICAL    CALCULATIONS. 

A  scientific  method  of  telling  imme- 
diately what  day  of  the  week  any  date 
transpired  or  will  transpire,  from 
the  commencement  of  the  Christian 
Era,  for  the  term  of  three  thousand 
years. 

MONTHLY   TABLE. 

The  ratio   to   add    for  each  month 
will  be  found  in  the  following  table : 


Ratio  of  June  is 0 

Ratio  of  September  is.. .  .1 

Ratio  of  December-  is 1 

Ratio  of  April  is 2 

Ratio  of  July  is 2 

Ratio  of  January  is 3 


Ratio  of  October  is 3 

Ratio  of  May  is 4 

Ratio  of  August  is 5 

Ratio  of  March  is 6 

Ratio  of  February  is G 

Ratio  of  November  is 6 


NOTE.  —  On  Leap  Year  the  ratio 
of  January  is  2,  and  the  ratio  of  Feb- 
ruary is  5.  The  ratio  of  the  other 
ten  months  do  not  change  on  Leap 
Years, 

CENTENNIAL    TABLE. 

The  ratio  to  add  for  each  century 
will  be  found  in  the  following  table : 

I 

fl     200,      900,    1800,    2200,     2600,    3000,  ratio  is 0 

3     300,    1000,     ratio  is 6 

•|     400,    1100,    1900,    2300,    2700 ratio  is 5 

0     500,    1200,    1600,    2000,    2400,    2800,  ratio  is 4 

600,     1300,       ratio  is 3 

000,    700,    1400,    1700,    2100,2500,    2900  ratio  is 2 

100,      800,    1500 ratiois 1 

NOTE.  —  The  figure  opposite  each 
century  is  its  ratio ;  thus  the  ratio  for 
200,  900,  etc.,  is  0.  To  find  the  ra- 
tio of  any  century,  first  find  the  cen- 
tury in  the  above  table,  then  run  the 
eye  along  the  line  until  you  arrive  at 
the  end,  the  small  figure  at  the  end  is 
its  ratio, 

METHOD*  OF   OPERATION. 

RULE.*  —  To  the  given  year  add 
its  fourth  part,  rejecting  the  fractions; 


*When  dividing  the  year  by  4,  always  leave  off  the 
centuries.  We  divide  by  4  to  find  the  number  <>f  Leap 
Tears. 


to  this  sum  add  the  day  of  the  month; 
then  add  the  ratio  of  the  month  and  the 
ratio  of  the  century.  Divide  this  sum 
by  7;  the  remainder  is  the  day  of  the 
week  counting  Sunday  as  the  first, 
Monday  as  the  second,  Tuesday  as  the 
third,  Wednesday  as  the' fourth,  Thurs- 
day as  the  fifth,  Friday  as  the  Sixth, 
Saturday  as  the  seventh  ;  the  remainder 
for  Saturday  will  be  O  or  zero. 

EXAMPLE  1. — Required  the  day  of  the 
week  for  the  4th  of  July,  1810. 

To  the  given  year,  which  is 10 

Add  its  fourth  part,  rejecting  fractions 2 

Now  add  the  day  of  the  month,  which  is 4 

Now  add  the  ratio  of  July,  which  is 2 

Now  add  the  ratio  of  1800,  which  is. 0 

Divide  the  whole  sum  by  7  7  |  18-4 

2 

We  have  4  for  a  remainder,  which 
signifies  the  fourth -day  of  the  week,  or 
Wednesday. 

Eule  for  finding  the  number  of  feet 
of  boards  which  can  be  cut  from  any 
log  whatever. 

From  the  diameter  of  the  log,  in 
inches,  substfact  4  for  the  slabs  and 
saw-calf.  Then  multiply  the  remainder 
by  half  itself  and  the  product  by  the 
length  of  the  log  in  feet,  and  divide 
the  result  by  8;  the  quotient  will  be 
the  number  of  square  feet. 

EXAMPLE  1. — What  is  the  number  of 
feet  of  boards  which  can  be  cut  from  a 
log  24  inches  in  diameter  and  12  feet 
long  ? 

Diameter 24  inches 

For  slabs  and  saw-call 4 

Remainder 20 

Half  remainder 10 

200 
Length  of  log 12 


300  the  number  of  feet. 


HENDERSON'S    LIGHTNING    PROCESS, 

For   Computing   Time  and  Interest,  Squaring  and  Multiplying  Numbers,  and  a 

Fine  Method  for  Dividing   Fractions,    and  an  infinite  number  of 

of  ways  of  Extracting   Square  and   Cube  Root. 


The  following  Table  gives  the  Interest  on   any  amount  at  7  per   cent. ,  by 
simply  removing  the  point  to  right  or  left,  as  the  case  may  require  : 


Number 
of  Days. 

$100 

$90 

$80 

$ro 

$60 

$50 

$40 

$30 

$30 

1. 

.0192 

.01726 

.01534 

.01342 

.01151 

.00950 

.00767 

.00575 

.00384 

2.  .. 

.0384 

.03452 

.03058 

.02685 

.02301 

.01918 

.01534 

.01151 

.00767 

3.  .. 

.0575 

.05178 

.04603 

.04027 

.03452 

.02877 

.02301 

.01726 

.01151 

4.  . 

.0767 

.06904 

.06137 

.05370 

.04603 

.02836 

.03068 

.02301 

.01536 

5.  . 

.0959 

.08630 

.07671 

.06712 

.05753 

.04795 

.03836 

.02877 

.01918 

6.  . 

.1151 

.10356 

.09205 

.08055 

.06904 

.05753 

.04603 

.03452 

.02313 

7.  . 

.1342 

.12082 

.  10740 

.09897 

.08055 

.06712 

.05370 

.04027 

.02685 

8.  . 

.1532 

.13808 

.12274 

.10740 

.09205 

.07671 

.06137 

.04603 

.03068 

9.  .  . 

.1726 

.15534 

.13808 

.  12089 

1.0356 

.08630 

.06904 

.05178 

.03452 

90. 

1.7260 

1.5342 

1.38082 

1.20822 

1.03562 

.86301 

.69041 

.51781 

.34521 

93.  .  . 

1.7836 

1.60521 

1.42685 

1.24849 

1.07014 

.89178 

.71342 

.53508 

.35671 

100...  . 

1.9178 

1.82603 

1.53425 

1.24247 

1.15065 

.95890 

.76712 

.57534 

.48356 

For  10-7  of  a  year  remove  the  decimal  point  one  place  to  the  left;  1-7,  or 
52  days,  two  places  to  the  left.  Increase  or  diminish  the  results  to  suit  the 
time, 

When  the  Eate  is  6  per  cent. 

For  5-3  of  a  year,  or  20  months,  remove  the  point  one  place  to  the  left;  60 
days,  two  places,  and  6  days  three  places  to  the  left. 


$5. 

7. 
8. 
9. 


00  $  94|7.50 

50.25  3458.50 

36.50  943.20 

47.75  Is  the  interest  at  7  per  64|9.3t) 

cent,  for  52  days,  or 

1-7  of  a  year. 


At  6  per  cent,  for  20 months; 
for  60  days  draw  the  line 
two  places  to  the  left  of 
the  decimal  point;  and  for 
6  days  three  places,  etc. 

When  the  rate  is  5  per  cent. 

For  two  years  remove  the  point  one  place  to  the  left,  and  73  days  two  places 
to  the  left. 

When  the  rate  is  7J  per  cent. 

For  4-3  of   a  year  or  16  months  remove  the  point  one  place  ;  for  48  days 
two  places,  the  result  modifying  to  suit  fhe  time  given. 


15 

When  the  rate  is  8  per  cent. 

For  15  months,  remove  the  point  one  place  ;  for  1-8  of  a  year,  or  45  days, 
two  places  to  the  left. 

To  MAKE  a  rule  for  all  rates,  divide  100  by  the  rate  and  the  quotient  is  the 
time,  when  the  principal  equals  the  interest  and  the  point  remains  the  same  ; 
divide  10  by  the  rate,  and  the  quotient  indicates  the  time  or  base  you  work  from, 
when  you  remove  the  point  1  place  to  the  left ;  divide  unity  by  the  rate,  and 
the  result  is  the  part  of  a  year  and  the  number  of  days,  when  the  point  is  to  be 
removed  two  places  to  the  left. 

To  FIND  THE  INTEREST  by  the  table,  for  any  given  time  and  any  number  of 
dollars,  look  on  the  Time  Table  for  the  time,  and  on  the  Interest  Table  for  the 
interest  of  twenty,  thirty  and  forty  dollars,  etc.  Modify  by  removing  the  point 
right  or  left  to  suit  the  example  given. 

You  can  find  the  interest  very  conveniently  by  taking  the  number  of  months 
and  J  of  the  days,  and  multiply  that  by  %  of  the  principal,  and  you  have  the 
interest  at  6  per  cent,  in  cents. 

RULE. — Eemove  the  point  one  place  to  the  left,  because  one-tenth  of  the 
principal  equals  the  interest. 

Remove  the  point  two  places,  for  one-hundredth  of  the  principal  equals  the 
interest. 

Remove  the  point  three  places,  because  one-thousandth  of  the  principal 
equals  the  interest. 

These  methods  give  the  interest  of  all  finite  sums  of  money,  for  the  time 
and  rate  mentioned  in  each  rule. 

To  reach  all  other  time,  increase  or  diminish  the  results  to  suit  the  time 
given. 

Thus:  $500  for  1-7  of  a  year  "at  7  per  cent,  is  five  dollars  ;  for  one  half  of 
that  time  $2.50  ;  for  one  fourth,  $1.25,  &c.;  for  one  year  it  is  seven  times  $5.00, 
$35. 

$400,  for  1-6  of  a  year,  and  rate  6,  is  $4  ;  for  one-half  of  that  time, 
$2. 

For  one  year  $24  ;  for  1-60  of  a  year,  or  6  days,  remove  the  point  three 
places  and  the  interest  is  40  cents  ;  for  one  half  of  that  time  it  is  20  cents. 

The  rule  may  thus  be  expressed  :  The  reciprocal  of  the  rate  is  the  time 
when  the  point  can  be  removed  two  places  to  the  left  in  all  cases  ;  ten  times 
that  time  remove  it  one  place  to  the  left,  one  tenth  of  the  same  time  three 
places  to  the  left  :  Increase  or  diminish  the  results  to  suit  the  time -given. 

TO   MULTIPLY   NUMBERS,    FIRST   KNOW   HOW   TO   SQUARE   THEM. 

(99)=9801  Take  the  comple-  (101)2=10201  When  above  the  base,  add  the  (11)2=121 
ment  of  99  from  it,  call  (102)2=10404  supplement,  call  it  hundreds,  and  (12)2=144 
it  hundreds,  and  add  the  (103)2=10609  increase  it  by  the  square  of  the  (13)2=169 
square  of  the  comple-  &c.,  &c.  supplement,  &c.,  &c. 

ment: 
Then  n=99 
and  c=l 
n+  c=100 

n_c=98 
n2-c2=9800 


16 


Now  add  c2  to  both  members  of  the  equation,  and  we  have  the  square  of  the 
number.  In  the  same  manner,  let  n  equal  the  number  and  s  the  supple- 
ment, and  the  reason  of  the  rule  becomes  evident.  For  same  reason: 

(98)2=9604 

(97)2=9409  Take  any  number  that  is  easy  to  multiply  by  for  the 

(96)2=9216  base  10,  20,  40,  50,  &c. 

(95)2=9025 
&c.,  &c. 

The  product  of  any  two  numbers  is  the  Square  of  the  Mean  diminished  by  the 
Square  of  Half  the  Difference.    . 


39  X41=(40)2— 12=1599 

38X42=(40)2— 22=1596 

37  X43=(40)2— 32=1591 

&c.,  &c. 

79X81=6399 
78X82=6396 


23X27=821 

22X26=616 

24X26=624 

&c.,  &c. 


From  the  square  of  the  mean  subtract  the  right 
hand  digit  of  the  greater  number  ;  because  it  in- 
dicates half  of  the  difference  of  the  two  numbers. 


8J    33    Multiply  both  dividend  and  divisor 
—  =  —        by  the  least  common  multiple  of 
6J     26        the  denominators  of  the  fraction- 
al parts. 


.  Increase  2  by  1,  and  multiply 
by  the  other  tens  digit,  and  an- 
nex the  product  of  unit's  digits. 
Add  1,  because  the  sum  of  the 
units  digits  is  =10. 


8  1-2X8  1-2=72  1-4 
8  1-3X8  2-3=72  2-9 
8  2-5X8  3-5=72  6-25 


c^  for  all  similar  ex- 
amples. 


HENDEKSON'S  METHOD  OF  EXTRACTING-  CUBE  BOOT. 


10000 

30000 

6000 

400 

36400 

6000 

800 

43200 

1800 

25 

45025 


1953125(100+20+5 
1000000 


953125 

728000 

225125 
225125 


Add  to  each  true  divisor,  as  they  occur,  twice 
the  surface  of  one  side  of  the  small  cube,  and  one 
of  each  of  the  three  parallelopipedons,  for  a  trial 
divisor;  because  that  will  make  three  sides  of  the 
complete  cube. 

By  observation  the  reason  is  evident  and  the 
conclusion  just,  for  making  trial  and  true  divisors 
by  this  method. 

We  have  an  infinite  number  of  ways  of  finding 
the  square  root,  cube  root,  &c.  a  Presume  the 
root  to  be  divided  into  a  certain  number  of  parts. 
Square  the  parts  in  square  root;  cube  them  in  cube 
root  to  find  the  divisor.  Thus  let  a-\-a  represent 
the  square  root  of  any  number.  The  square  of 
a-\-a  is  4  a2 :  hence  divide  any  number  by  4  and 
extract  the  square  root  of  the  quotient,  and  we 
have  half  of  the  root.  Divide  any  number  by  the 
square  of  3,  and  extract  the  square  root  of  the 
quotient,  and  we  have  one-third  of  the  root,  &c., 
for  all  numbers.  In  the  cube  root  we  cube  the 
number  representing  the  parts  the  root  is  divided 
into,  for  a  divisor 


17 


To  find  the  Day-of  the- Week  from  the  Day  -of  the-Month. 

Cast  the  sevens  out  of  the  day  of  the  month,  the  ratio  of  the  month,  the 
ratio  of  the  year,  and  the  year.  One  of  a  remainder  will  be  the  first  day  of  the 
week ;  two  second,  &c.,  0  the  last  day  of  the  week.  The  ratio  of  the  month  is 
found  above  its  name.  The  ratio  of  every  month  except  January  and  February 
is  one  more  in  Leap  Years, 


Jan'y 
3 

Feb'y 
6 

March 
6 

April 
2 

May 

June 
0 

July 

2 

August.   Sept. 
5      -1  — 

October 
3 

Novem. 
6 

Decem. 
j 

1   1 

1  32 

1  60 

1  91 

1  121 

1  152 

1  182 

1  213  !  1  244 

1  274 

1  305 

1  335 

2  2 

2  33 

2  61 

2  92 

2  122 

2  153 

2  183 

2  214!  2  245 

2  275 

2  306 

2  336 

3  3 

3  34 

3  62 

3  93 

3  123 

3  154 

3  184 

3  215  3  246 

3  276 

3  307 

3  337 

4  4 

4  35 

4  63 

4  94 

4  124 

4  155 

4  185 

4  216  4  247 

4  277 

4  308 

4  338 

5  5 

5  36 

5  64 

5  95 

5  125 

5  156 

5  186 

5-  217 

5  248 

5  278 

5  309 

5  339 

6  6 

6  37 

6  65 

6  96 

6  126 

6  157 

6  187 

6  218 

6  249 

6  279 

6  310 

6  340 

7  7 

7  38 

7  66 

7  97 

7  127 

7  158 

7  188 

7  219 

7  250 

7  280 

7  311 

7  341 

8  8 

8  39 

8  67 

8  98 

8  128 

8  159 

8  189 

8  220 

8  251 

8  281 

8  312 

8  342 

9  9 

9  40 

9  68 

9  99 

9  129 

9  160 

9  190 

9  221 

9  252 

9  282 

9  313 

9  343 

10  10 

10  41 

10  69 

10  '100 

10  130 

10  161 

10  191 

10  222 

10  253 

10  283 

10  314 

10  344 

11  11 

11  42 

11  70 

11  101 

11  131 

11  162 

11  192 

11  223 

11  254 

11  284 

11  315 

11  345 

12  12 

12  43 

12  71 

12  102 

12  132 

12  163 

12  193 

12  224 

12  255 

12  285 

12  316 

12  346 

13  13 

13  44 

13  72 

13  103 

13  133 

13  164 

13  194 

13  225 

13  256 

13  286 

13  317 

13  347 

14  14 

14  45 

14  73 

14  104 

14  134 

14  165 

14  195 

14  226 

14  257 

14  287 

14  318 

14  348 

15  15 

15  46 

15  74 

15  105 

15  135 

15  166 

15  196 

15  227 

15  258 

15  288 

15  319 

15  349 

16  16 

16  47 

16  75 

16  106 

16  136 

16  167 

16  197 

16  228 

16  259 

16  289 

16  320 

16  350 

17  17 

17  48 

17  76 

17  107 

17  137 

17  168 

17  198 

17  229 

17  260 

17  290 

17  321 

17  351 

18  18 

18  49 

18  77 

18  108 

18  138 

18  169 

18  199 

18  230 

18  261 

18  291 

18  322 

18  352 

19  19 

19  50 

19  78 

19  109 

19  13* 

19  170 

19  200 

19  231 

19  262 

19  292 

19  323 

19  353 

20  20 

20  51 

20  79 

20  110 

20  140 

20  171 

20  201 

20  232 

20  263 

20  293 

20  324 

20  354 

21  21 

21  52 

21  80 

21  111 

21  141 

21  172 

21  202 

21  233 

21  264 

21  294 

21  325 

21  355 

22  22 

22  53 

22  81 

22  112 

22  142 

22  173 

22  203 

22  234 

22  265 

22  295 

22  326 

22  356 

23  23 

23  54 

23  82 

23  113 

23  143 

23  174 

23  204 

23  235 

23  266 

23  296 

23  327 

23  357 

24  24 

24  55 

24  83 

24  114 

24  144 

24  175 

24  205 

24  236 

24  267 

24  297 

24  328 

24  358 

25  25 

25  56 

25  84 

25  115 

25  145 

25  176 

25  206 

25  237 

25  268 

25  298 

25  329 

25  359 

26  26 

'26  57 

26  85 

26  116 

26  146 

26  177 

26  207 

26  238 

26  269 

26  299 

26  330 

26  360 

27  27 

27  58 

27  86 

27  117 

27  147 

27  178 

27  208 

27  239 

27  270 

27  300 

27  331 

27  361 

28  28 

28  59 

28  87 

28  118 

28  148 

28  179 

28  209 

28  240 

28  271 

28  301 

28  332 

28  362 

29  29 

29  88 

29  119 

29  149 

29  180 

29  210 

29  241 

29  272 

29  302 

29  333 

29  363 

30  30 

30  89 

30  120 

30  150 

30  181 

30  211 

30  242 

30  273 

30  303 

30  334 

30  864 

31  31 

31  90 

31  151 

31  212 

31  243 

31  304 

31  365 

SUGGESTIONS  ON  TEACHING  ARITHMETIC. 


Qualifications. — The  chief  qualifica- 
tions requisite  in  teaching  Arithmetic, 
as  well  as  other  branches,  are  the  fol- 
lowing:— A  thorough  knowledge  of  the 
subject;  a  love  for  th<!  employment; 
and  an  aptitude  to  teach.  These  are 
indispensable  to  success. 

Classification  — Arithmetic,  as  well 
as  other  studies,  should  be  taught  in 
classes.  This  method  saves  much  time, 
and  thereby  enables  the  teacher  to  de- 
vote more  time  to  oral  illustrations. 

The  action  of  mind  upon  mind  is  a  po- 
tential stimulant  to  exertion,  and  can- 
not fail  to  create  a  zeal  for  the  study. 
The  mode  of  analyzing  and  reasoning 
of  one  scholar  often  suggests  new  ideas 
to  others  in  the  class. 

In  classification,  those  should  be  put 
together  who  possess  as  nearly  equal 
capacities  as  possible.  If  any  of  the 
class  learn  faster  than  the  others,  they 
should  be  allowed  to  take  extra  study, 
or  be  furnished  with  additional  exam- 
ples to  solve,  so  that  the  whole  class 
may  advance  together. 

The  Blackboard  should  be  one  of  the 
indispensables  of  the  school-room.  Not 
a  recitation  should  pass  without  its  use. 
When  a  principle  is  to  be  demonstrated 
or  an  operation  explained,  if  done  upon 
the  blackboard, 'all  can  see  and  under- 
stand at  once. 

Recitation. — The  first  object  in  con- 
ducting a  recitation  should  be  to  se- 
cure the  attention  of  the  class.  This  is 
done  chiefly  by  blending  life  and  variety 
with  the  exercise.  Students  generally 
loathe  dullness,  while  animation  and 
variety  are  their  delight.  Every  exam- 
ple should  be  carefully  analyzed;  the 
"why  and  wherefore"  of  every  step  in 
the  solution  should  be  required,  till  the 
learner  becomes  perfectly  familiar  with 
the  process  of  reasoning. 

Thoroughness. — This  should  be  the 
motto  of  every  teacher;  without  it,  the 
great  objects  of  study  are  radically  de- 
feated. In  securing  this  object,  much 
advantage  is  derived  from  frequent  re- 
views. 


AKITHMETIC. 

Arithmetic  is  the  science  of  numbers, 
and  the  art  of  computing  by  them. 

A  number  is  a  unit  or  a  collection  of 
units. 

A  unit  is  a  single  thing,  or  one. 

Quantity  is  anything  that  can  be  in- 
creased, diminished,  or  measured. 

The  fundamental  rules  of  Arithmetic 
are  Addition,  Subtraction,  Multiplication, 
and  Division. 

NUMERATION. 

Numeration  is  the  process  of  reading 
numbers  when  expressed  by  figures. 

Figures  are  characters  used  to  express 
numbers  in  arithmetic.  There  are  ten: 
1,  2/3,  4,  5,  6,  7,  8,  9,  0.  Each  figure 
has  two  values — simple  and  local.  The 
simple  value  is  that  expressed  by  the 
figure  when  standing  alone  or  in  the 
unit's  place. 

The  local  value  is  that  expressed  when 
connected  with  other  figures,  and  de- 
pends upon  its  distance  from  the  unit's 
place. 

The  cipher  denotes  the  absence  of 
something,  and  when  placed  to  the  right 
of  a  figure,  it  increases  the  value  of  that 
figure  ten  times,  or  multiplies  it  by  ten. 

The  value  of  a  figure  is  illustrated  by 
the  following  numeration  table: — 


ill 

w  H  S 
321 


111 

321 


This  table  may  be  run  on  to  Quadril- 
lions, Quintillions,  Sextillions,  Septil- 
lions,  Octillions,  Nonillions,  Decillions, 
Undecillions,  Duodecillions,  Tredecil- 
lions,Quatuordecillions,  Quindecillions, 
Sexdecillions,  Sepdecillions,  Ocdecil- 
lions,  Nondecillions,  Vigintillions,  etc. 


APPENDIX  TO  LIGHTNING  CALCULATOR, 

CONTAINING  GENERAL  INFORMATION. 


ARITHMETICAL  SIGNS. 

—Sign  of  equality  5  =  5,  read  five 
equals  five. 

-|-  Plus,  the  sign  of  addition;  4+8 
=12,  read  four  plus  eight  equal  twelve. 

—  Minus,  the  sign  of  subtraction;  9 — 
3  —  6,  read  nine  minus  three  equals  six. 
X  Sign  of  multiplication;  4  X  2=  8, 
read  four  multiplied  by  two  equals  eight. 

-j-  Sign  of  division;  12-^-3—4,  read 
twelve  divided  by  three  equal  four. 

I/  Eadical  sign,  or  sign  of  square 
root,  when  placed  over  a  number,  signi- 
fies that  the  square  root  is  to  be  ex- 
tracted; |/  64=8,  read  the  square  root 
of  sixty-four  equals  eight. 

^  Sign  of  cube  root,  shows  that  the 
cube  root  is  to  be  extracted;  ^  27=3, 
read  the  cube  root  of  twenty-seven 
equals  three. 

SHOKT  METHODS  OF  MULTIPLI- 
CATION AND  DIVISION 

To  multiply  by  25. 
1.  Multiply  492  by  25. 

Operation. 
4149200 

Analysis. — By  annex-  112300  Ans. 
ing  two  ciphers  to  the  multiplicand, 
we  multiply  it  by  100,  which  gives  a 
product  four  times  too  great,  as  the 
multiplier,  25,  is  but  one-fourth  of  100. 
Hence,  annex  two  ciphers  to  the  mul- 
tiplicand, and  divide  by  4. 


2.  What  will  45  acres  of  land  cost  at 
$25  an  acre  ? 

Ans.  $1125. 
To  multiply  by  12J. 

Annex  two  ciphers  and  divide  by  8. 
1.  Multiply  38  by  12 J.   Operation 


8380.0 


Analysis.  — Annexing 
two  ciphers  in  this  case 
produces  a  product  eight 


475   Ans. 

times  too 
great;  hence  divide  by  8  to  obtain  one- 
eighth. 

2.  What  will  13  yards  calico  cost  at 
12^  cents  per  yard?  Ans.  $1.62J. 

3.  Multiply  12£  by  143.    Ans.  1787^ 

To  multiply  by  33£ 

Annex  two  ciphers  and  divide  by  3. 

1.  What  will  27   yards  cloth   cost  at 
33  J  cents  per  yard?  Ans.  $9. 

2.  Multiply  33  J  by  14.         Ans.  466§. 

3.  Multiply  47  by  33J.       Ans.  1566§. 

To  multiply  by  125. 
Annex  three  ciphers  and  divide  by  8. 

1.  Multiply  568  by  125, 

Operation. 
8  568000 

Analysis. — The  stu-       

dent  will  readily  per-  71000  Ans. 
ceive  that  by  annexing  three  ciphers  he 
multiplies  by  1000;  and  that  125  is  one- 
eighth  of  1000.  Hence  divide  by  8. 

2.  What  will  44  pair  of  shoes   cost 
at  $1.25  per  pair?  Ans.  $55.00. 

3.  What  will  125  yards  cloth  cost  at 
58  cents  per  yard?  Ans.  $72.50. 


20 


To  multiply  any  number  ending  in  5  by 
itself;  that  is,  to  square  such  a  number. 
1.  Multiply  25  by  25.  Operation. 

25 
25 

625  Ans. 

RULE. — Square  the  5  and  set  down 
the  result,  25,  then  multiply  the  left-hand 
figure  by  the  next  figure  above  it  in  the 
order  of  numbers  and  prefix  the  product 
to  the  25,  and  you  have  the  correct  result. 

NOTE.— This  rule  is  based  upon  the  well-known  prin- 
ciple of  squaring  any  number  consisting  of  tens  and 
units.  The  square  of  such  a  number  is  equal  to  the 
square  of  the  tens,  plus  twice  the  product  of  the  tens 
by  the  units,  plus  the  square  of  the  units. 

Thus  25  is  composed  of  2  tens  or  20 
and  5  units;  and  the  20  squared  equals, 

400 

20  multiplied  by  5  equals- 100,  twice 
that  is  200 

And  lastly  ^  the  5  squared  equals         25 


625 

Again,  5  squared  is  25,  and  the  2 
multiplied  by  3  (the  next  figure -above 
it)  is  6,  which  placed  before  the  25 
makes  625e 

Operation. 

2.  Multiply  45  by  45,    Or 
thus: —  45 

45 

5  squared  equals  25,  and       

five  times  4  equals  20,  and  20       225 
prefixed  to  25  makes  2025.,     180 
Ans. 

2025 

3.  Square  85.  Ans.  7225. 

4.  What  will  65  yard& of  cloth  cost  at 
65  cents  per  yard  ?  Ans.  42.25. 

5.  What  shall  I  pay  for  75  acres  of 
land  at  $75  per  acre  ?  Ans..  $5625. 
To  multiply  a  mixed  number  -ending  in  \ 

by  itself < 

RULE. — Multiply  the  figure  or  number 
by  the  next  figure  above  it,  and  place  the 
square  of  J  (which  is  \  always)  to  the 
right. 


1.  Multiply  2J  by  2£.  Ans.  6J. 
Operation.  —  The   2   multiplied  by   3 

is  6,  and  a  half  times  a  half  is  J;  hence 
6J  is  the  product. 

2.  Multiply  4J  by  4J.          Ans.  20J. 
Say  five  times  4  is  20  and  annex  the  J. 

3.  What  will  12  J  yards  of  cloth  cost 
at  12J  cents  per  yard?         Ans.  $1.56J. 

Say  13  times  12.  So  of  any  other 
numbers. 

To  square  any  mixed  number  that  ter- 
minates with  J, 

RULE.  —  Square  the  J;  then  square  the 
whole  number,  add  half  the  whole  number 
is  its  square,  and  place  the  result  to  the 
left  of  the  TV 

1.  What  is  the  square  of  4J?  Ans.  18T1F. 
Operation.  —  J     squared    is    TV  ;     4 

squared   is  16  and  2  added  makes  18. 
Then  18TV 

2.  What  is  the  -square  of  £J? 

'Ans. 

3.  What  is  the  square-  of  12|? 

Ans. 

4.  What  is  the-  square-  of  -8J  ? 

Ans. 


SHORT  METHODS   OF  DIVISION. 

To  divide  by  25. 

1.  Divide-425  by  25,  Ans,  17. 

Operation. 
425 


Analysis. — By  multi- 
plying both  divisor  and 
dividend  by  4,  the  divi- 
sor becomes  100,  which 
enables  us  to  perform 
the  division  by  simply 
cutting  off  two  figures 
from  the  right. 


17.00 

or 
25=     425 

4         4 


1.00 


17.00 


17  Ans. 


RULE.  —  Multiply  both  divisor  and 
dividend  iy  that  number  which  will  change 
the  divisor  to  a  number  of  tens,  hundreds, 
or  thousands,  and  then  divide  by  simply 
cutting  off  figures  from  the  right. 


21 


2.  Divide  3489  by  25. 

Ans.  139.56  or  £f . 

Simply  multiply   by  8  and  point  off 
two,  as  12 J  is  one  eighth  of  100. 

3.  Divide  4800  by  12J          Ans.  384. 

4.  Divide  54500  by  125.         Ans.  436. 
Multiply  by  8  and  cut  of  three. 

5.  Divide  5470  by  250. 

Ans.  21.880  or  21ff. 
Simply   multiply   by  4  and  point  off 
three  figures,  and  to  the  right  is  thou- 
sandths, 

6.  Divide  2000  by  333J  Ans.  6. 
Multiply  by  3  and  cut  off  three,  it 

being  the  third  of  a  thousand.     And  so 
of  other  numbers. 

7.  Divide  $1400  by  33£.          Ans.  42. 

8.  Divide  155  by  33 J.  Ans.  4.65. 

9.  Divide  1500  by  33£.  Ans.  45. 


PEACTICAL    PROBLEMS  ANA- 
LYZED, 

1.  Divide  $140  among  three  boys,  B 
to  have  twice  as  much  as  A,  and  C  twice 
as  much  as  B. 

Ans.  A  20,  B  40,  and  C  80. 
Operation. 

1  1   X  20  =  20  A 

2  2  X  20  =  40  B 
4                    4  X  20  =  80  G 

7140 


If 


20 


A  gets  1  dollar,  B  2,  and  C.  4,— 
all  together  they  will  get  7  dollars; 
but  7  in  140  twenty  times,  therefore  each 
will  get  20  times  the  "index  number; 
hence  multiply  each  by  20. 

2.  If  J.  W.  Butler  sells  three  pairs  of 
shoes  for  $12.33;  the  second  twice  as 
much  as  the  first,  and  the  third  twice  as 
much  as  both  the  others;  what  does  he 
get  for  each  pair  ? 

Ans.  $1.37,  $2.74,  $8.22. 

3.  If  I  have  f  of  my  money  in  one 
pocket,  |  in  a  second,  ai.d  eight  dollars 
in  a  third,  how  much  money  have  I  ? 

Ans.  $180. 


Operation.—  f  -f  f  =*§;  1— «=*. 

Then,  8  divided  by  4^  =  180,  Ans. 

In  such  examples  we  merely  get  the 
sum  of  the  fractional  parts,  substract  it 
from  1,  and  divide  the  given  number  by 
the  fractional  remainder. 

4.  E.  P.    Williams  has  his  goats  in 
five  fields,  in  the  first  he  had  J  of  them, 
in  the   second  i,  in  the  third  J,  in  the 
fourth   T*2,  and  in  the  fifth  225,   how 
many  had  he  ?  Ans.  600. 

5.  John  Butler  leaves  his  son  an  es- 
tate, J  of  wyhich  he  spends  in  5  months, 
f  of  the  remainder  in  10  months,  and 
then  had  $500  left,  what  was  the  estate  ? 

Ans.  $3000. 

Operation. — 1 — J=§;    1 — |=J 
Then  JXiRi;  500-v-J= 

Ans.  $3000. 

After  spending  £  he  has  f  left.  After 
spending  f  of  that,  he  must  have  J  of 
the  f  left,  which  is  i  of  the  whole ; 
then  500  must  be  i  of  the  whole 

6.  From  a  drove  of  beeves  I  sell  to  A 
|,   to  B   §    of  the  remainder,  to  C   $ 
of  the  remainder,  and   have  50  beeves 
left;  how  many  at  first?  Ans.  450. 

7.  A  cistern  has  two  pipes  for  filling 
it, — one  in  6  hours  and  the  other  in  12 
hours ;  it  also  has  a  discharging  pipe, 
which   empties    it  in   5   hours.      Now 
leave  all  open,  in  what  time  will  the 
cistern  fill  ?  Ans.  20  hours. 

Operation.  —  i  -f  TV=  J ;  J — i  =  ?V 
1-^=20,  Ans. 

It  is  plain  that  the  two  filling  pipes 
will  fill  J  of  the  cistern  in  1  hour,  while 
the  discharging  pipe  will  empty  i  in  1 
hour.  Their  difference  shows  how  much 
is  filled  in  1  hour.  Then  1,  the  whole 
cistern,  divided  by  ^  will  give  the  hours. 

8.  A  water  tank  has  two  filling  pipes. 
The  first  would  fill  it  in  40  minutes,  the 
second  in  50.    It  has  a  discharging  pipe, 
which  empties  it  in  25  minutes.     Now 
suppose  all  turned  on  at  once,  how  long 
would  the  tank  be  in  filling? 

Ans.  3  hours  and  20  minutes. 


22 


9.  John  can  do  a  job  of  work  in  5 
days,   and  James  in  8  days;   in   what 
time  can  they  both  do^the  work  together? 

Ans.  3  -fg  days. 

Operation. — 8  X  5=40;  5-j-8  =  13; 
40  •+-  13  =  3TV,  Ans. 

We  may  consider  the  work  divided 
into  40  equal  parts  (or  any  other  com- 
mon multiple-of  5  and  8),  and  we  plainly 
see  that  John,  working  5  days,  will  do 
eight  of  those  parts  each  day,  and  James 
five;  both  together  will  do  in  1  day  13 
parts;  hence  it  will  require  as  many 
days  as  13  is  contained  times  in  40. 

10.  Sallie  can  make  a  coat  in  6  days, 
Jane  in  4  days;  how  long  required  for 
both  to  make  it  together?  Ans.  2|  days. 

11.  A  can  do  a  piece  of  work  in  10 
days,  B  in  12  days,  and  C  in  15  days; 
in  what  -time  could   all  together  accom- 
plish it  ?  Ans.  4  days. 

12.  Three   men  can  do  a  job  in  9 
days.     A  alone~can  do  it  in  18  days,  B 
in  27  days;  in  what  time  can   C  do  it 
alone?  Ans.  54-days. 

Operation.— -27  —  1&=  9.  27  X  18 
-i-9  =  G4,  Ans. 

In  this  case  we  divide  any  common 
multiple  of  the  two  given  numbers  by 
their  difference,  and  the  quotient  shows 
the  other  number, 

13.  A  man  and  his  wife  together  can 
drink  a  demijohn  of  whisky  in  12  days, 
but  when  the  "old  man"  is  absent  it 
lasts  the  old  lady  30 -days.     Now  if  the 
old  lady  leaves  home,  how  long  will  it 
last  the  old  man  ?  Ans.  20  days. 
14.  Which  will  enclose  the  most  ground , 

A  fence  made  square,  or  one  made 

round, 

Two  panels  to  each  rod  of  land, 
Ten  rails  in  each,  we  understand; 
And  every  rail  in  each  suppose 
To  just  one  acre  of  land  enclose; 
The  next  thing  is  to  tell  exact 
How  many  acres  in  each  tmct  ? 

A        ( 1024000  in  the  square. 
Ans'  \   804571f  in  the  circle. 


Operation.  —  r^  =  area  of  1  rod 
square,  which  takes  80  rails  to  fence  it. 

Then  T  JT  :  80  :  :  80  =  1024000  rails 
or  acres,  Ans. 

And  as  one  side  of  the  square  equals 
the  diameter  of  the  circle,  we  multiply 
1024000  by  |£  and  obtain  the  circle  =L 
804571^  rails  or  acres,  Ans. 

15.  A,  B  and  C  agree  to  grade  a  piece 
of  railroad.  A  and  B  can  do  the  work 
in  16  days,  B  and  C  in  13J  days,  and  A 
and  C  in  11^  days.  In  how  many  days 
can  all  do  it,  working  together,  and  in 
how  many  days  can  each  do  it  alone  ? 

Operation  . 

TV  =  wu,  what  A  and  B  do  in  1  day. 
?3o  =  /o,  what  B  and  C  do  in  1  day. 
/^  =  s7^,  what  A  and  C  do  in  1  day. 


=  IS 


8o        8p 
in  2  days. 

Jg  -f-  2  =  890,  what  all  will  do-  in  1  day. 
-i_  g9o  —  8|  days,  time  A,  B,  and  C 
will  do  the  whole  work  together. 


so 


_     5    

80 


8YC  alone. 


80 


=  830  A  alone. 
=8aoB  alone. 


And 

l-i-840  =  20daysforC. 
1  -4-  8«0  =  26§  days  for  A. 
l-=-820=40daysfor  B. 

Analysis. — Since  A  and  B  can  do  the 
work  in  16  days,  they  can  do  ^  of  it  in 

I  day;  B  and  C,  in  13 J  or  *3°  days,  they 
can  do  430  of   it  in  1  day;  A  and  C  in 

II  f,  or  87°  days,  they  can  do  870  in  1  day. 
Then  A,  B,  and  C,  in  two  days  can  do 
| I  of  the  work,  and  in  1  day  890 ;  and  it 
will  take  them  as  many  days  working 
together  to  do  the  whole  work  as  890  is 
contained  times  inl,  or  8|  days.    Now, 
if  we  take  what  any   two  do  in  1  day 
from  what  three  do  in  1  day,  the  re- 
mainder will  show  what  part  the  third 
does  in  1  day.     We   thus   find   that  A 
does  830,  B  8»0,  and  C  8*0, 

Next  denote  the  work  by  1  and  divide 
it  by  each  of  these  fractions,  and  the 
quotient  will  express  the  days  required 
by  each  to  do  it  by  himself. 


23 


16.  A  and  B  can  do  a  piece  of  work, 
in  5  fa  days,  B  and  C  in  6-f  ,  A  and  C 
in  6  days;  in  what  time  would  all  do 
the  work  together,  and  each  alone  ? 

Ans.  All,  4  days;  A,  10;  B,12;  C,  15. 


WONDEKS  OF  NUMERATION. 

By  the  tables,  as  given  in  most  of  our 
school  arithmetics,  numeration  is  car- 
ried only  to  six  places,  or  quadrillions, 
running  up  by  terms  derived  from  the 
Latin  numerals. 

A  series  of  units  of  that  extent  would 
be  beyond  the  power  of  man  to  compre- 
hend, or  even  imagine.  Even  millions 
convey  a  very  indefinite  idea,  and  when 
it  rises  to  billions,  the  mind  can  no 
longer  grasp  the  number;  for,  though  we 
may  read  the  expression,  it  is  very  much 
as  we  read  sentences  in  an  unknown 
language.  But  we  may  perhaps  assist 
the  mind  of  the  student  by  some  little 
calculation.  Often  do  we  see  millions 
spoken  of  in  our  national  expenditures; 
and  yet  even  that  is  an  exceedingly 
large  number,  for  if  a  man  were  to  count 
fifteen  hundred  dollars  an  hour,  and 
work  faithfully  eight  hours  a  day,  it 
would  take  him  nearly  three  months  to 
count  a  million  of  dollars;  and  if  the 
dollars  were  silver,  If  inches  in  diame- 
ter, laid  touching  each  other  in  a  straight 
line,  they  would  reach  over  twenty-five 
miles,  and  thirty-one  wagons,  hauling 
two  thousand  pounds  each  would  not  be 
sufficient  to  haul  them;  and  were  they 
greenback  dollar  bills  laid  in  this  line, 
they  would  form  a  line  over  one  hun- 
dred and  ten  miles  long. 

Our  only  plan,  then,  to  understand 
this,  is  to  group  the  number  by  imagin- 
ing one  thousand  piles  or  lots,  and  one 
thousand  dollars  in  each  pile,  when  we 
can  gain  as  distinct  an  idea  of  the  num- 
ber of  piles  as  of  the  individuals  of  each 
pile.  But  suppose  we  extend  even  this 
mode  to  such  a  sum  as  our  present  na- 


tional debt,  and  we  are  lost  in  wonder 
and  amazement,  and  the  mind  is  utterly 
bewildered.  The  present  national  debt 
is  about  three  billions,  and  there  are 
about  two  thousand  clerks  in  the  Treas- 
ury Department.  Now,  were  they  all 
to  turn  their  attention  to  counting  this 
money,  and  work  eight  hours  a  day 
counting  one  every  second,  it  would  re- 
quire between  three  and  four  months  to 
accomplish  the  work. 

Dr.  Thompson,  Professor  of  Mathe- 
matics, at  Belfast,  Ireland,  very  justly 
remarks :  ' '  Such  is  the  facility  with 
which  large  numbers  are  expressed, 
both  by  figures  and  in  language,  that 
we  generally  have  a  very  limited  and  in- 
adequate conception  of  their  real  mag- 
nitude. The  following  considerations 
may  perhaps  assist  in  enlarging  the 
ideas  of  the  pupil  on  this  subject: — 

"  To  count  a  million,  at  one  per  sec- 
tion, would  require  between  twenty- 
three  and  twenty-four  days,  of  twelve 
hours  each. 

' '  The  seconds  in  six  thousand  years 
are  less  than  one-fifth  of  a  trillion.  A 
quadrillion  of  leaves  of  paper,  each  the 
two-hundredth  part  of  an  inch  in  thick- 
ness, would  form  a  pile  the  height  of 
which  would  be  three  hundred  and 
twenty  times  the  distance  of  the  moon 
from  the  earth.  Let  it  also  be  remem- 
bered that  a  million  is  equal  to  a  thou- 
sand repeated  a  thousand  times,  and  a 
billion  equal  to  a  million  repeated  a 
thousand  times." 

A  rifle  ball  flies  twelve  hundred  feet 
per  second,  and  if  one  were  fired  at  the 
moment  one  of  the  presidents  of  the 
United  States  takes  his  seat,  and  con- 
tinued unabated  for  the  four  years,  it 
would  not  travel  three  millions  of  miles. 
Suppose  a  man  were  to  count  one  every 
second  of  time,  day  and  night,  without 
stopping  to  rest,  eat,  drink,  or  sleep,  it 
would  take  him  thirty-two  years  to  count 
a  billion,  and  thirty-two  thousand  years 
to  count  a  trillion.  What  a  limited  idea 


24 


we  generally-entertainof  the  immensity 
of  numbers! 


THE     MILLER  S   RULE     FOR    WEIGHING 
WHEAT. 

Wheat  weighing  58  pounds  and  up- 
wards per  bushel  is  considered  mer- 
chantable wheat,  and  60  pounds  of 
merchantable  wheat  make  a  standard 
bushel.  Hence,  wheat  weighing  less 
than  60  pounds  per  bushel  will  lone  in 
making  up;  but,  weighing  more,  it  will 
gain. 

When  wheat  weighs  less  than  58 
pounds  per  bushel,  it  is  customary,  on 
account  of  the  inferior  yield  of  light 
wheat,  to  take  two  pounds  for  one  in 
making  up  the  weight;  hence,  it  will 
take  63  pounds  to  make  up  a  bushel, 
provided  the  wheat  weighs  but  57,  and 
64  if  the  wheat  weighs  but  56  pounds 
per  bushel. 

CASE  I. — To  change  merchantable 
wheat  to  standard  weight. 

RULE. — Bring  the  whole  quantity  of 
wheat  to  pounds  and  divide  by  60. 

EXAMPLE  1. — How  many  standard 
bushels  of  wheat  are  in  150  bushels, 
each  weighing  58  pounds  ? 

150      Or,  each  bushel  lacks  2  Ibs. ;         150 
58  2 


1200 
750 

From  150  bush. 

6,0)870,0  Take      5 


6,0)30,0 
Deficiency,     5 


Ans.  145  b.  Leaves  145,  the  answer. 

2.  How  many  standard  bushels  of 
wheat  are  in  80  bushels  45  pounds, 
weighing  63  ? 


Bush.  Ibs. 
80  45 
63 

285 
480 

6,0)508,5 


Or,  80  bush. 
3 

6,0)24,0  excess  of  weight. 

4  bush. 
80        45  Ibs. 


Ans.  84  b.45  lbs.=84%b.  Ans.  84b.  451bs.  or  3p. 


3.  How  many  standard  bushels   oi 
wheat  are  in  175  bushels  37  pounds 
weighing  59?      Ans.  172  bush.  42.  Ibs. 

4.  How  many  standard  bushels  are  in 
100  bushels   15   pounds,  weighing   62 
pounds  per  bushel  ?    Ans.  103  bus.  35 
Ibs. 

CASE  II. — When  wheat  weighs  less 
than  58. 

RULE. — Bring  the  whole  quantity  to 
pounds,  and  divide  by  as  many  pounds  as 
make  a  standard  bushel  of  such  wheat. 

EXAMPLE  1. — How  many  bushels  of 
good  wheat  are  equal  to  100  bushels 
weighing  57? 

100  Or,  Gibs,  per  bus.=600  Ibs. 

57  63)600(9  bus.  33  Ibs.  defect. 

567. 

63)5700(90  bush.  30  Ibs. 

567.  From  100  bush.    33 

Take  9  33 

30  Ibs. 

Ans.  90  30 

NOTE. — The  odd  pounds  in  the  above 
and  following  results  are  also  subject 
to  a  small  drawback,  viz.,  1  Ib.  in  every 
21  when  the  vveight  weighs  57;  1  in  16 
when  it  weighs  56,  and  so  on;  conse- 
quently, the  above  ought,  in  strictness, 
to  be  90  bushels,  and  rather  more  than 
28J  pounds,  but  millers  seldom  make 
this  deduction. 

2.  How  many  standard  bushels  of 
merchantable  wheat  will  be  equal  to 
250  bushels  18  Ibs.  weighing  56  Ibs.  per 
bushel  ?  Ans.  219  bush.  2  Ibs. 

How  much  good  wheat  is  equal  to 
1000  bushels  weighing  55  ?  Ans.  846 
bush.  10  Ibs. 

NOTE. — before  dismissing  this  rule  it 
appears  proper  that  a  few  remarks 
should  be  made,  in  order  to  show  the 
young  farmer  the  importance  of  under- 
standing it  properly.  There  are  dif- 
ferent methods  of  '  'making  up  wheat" 
(i.  e.,  finding  its  merchantable  value), 
and  these  methods  give  different  re- 
sults; hence  the  necessity  of  the  subject 
being  understood  by  all  concerned.  I 
shall  not  undertake  to  determine  be- 
tween the  farmer  and  the  miller  which 


25 


is  or  which  is  not  the  fair  way;  but, 
after  explaining1  the  principle,  leave 
them  to  make  their  bargains  as  they 
may  choose. 

If  I  have  a  bushel  of  wheat  that  weighs 
but  57  pounds,  then  six  pounds  of  the 
same  kind  of  wheat  will  be  necessary  to 
make  this  a  merchantable  bushel,  so 
that  63  pounds  of  this  quality  of  wheat 
will  make  a  standard  bushel;  and  it  is 
upon  this  supposition  that  the  preced- 
ing calculations  are  founded.  But  a 
number  of  millers  use  a  method  of  cal- 
culation by  which  they  take  good  wheat 
for  the  odd  pounds — i.  e. ,  they  take  a 
bushel  full,  say  57  pounds,  of  the  .wheat 
they  are  measuring,  and  instead  of 
taking  six  pounds  more  of  the  same  kind 
to  make  it  up,  they  take  six  pounds  of 
good  or  merchantable  wheat.  Their 
method  of  calculation  is  as  follows : — 

Kequired,  the  good  wheat  in  1000 
bushels  weighing  55  ? 


Defect,  10  Ibs.  per  bush.  1000 


From  1000  bush. 
Take     1G6  40 

Ans.  833  20 


10 


6,0)1000,0 


Bush.  166  40 


We  see  that  this  gives  a  result  nearly 
13  bushels  more  in  the  miller's  favor 
than  the  former  method;  and  this  I 
know  to  be  practiced  by  many, 

*^ 

There  is  another  method   sometimes 

used  by  those  who  are  not  very  scrupu- 
lous in  their  distinctions  between  right 
and  wrong.  They  find  the  whole  defect 
in  pounds,  and  divide  by  the  weight  of 
a  bushel  of  the  wheat  to  find  how  many 
bushels  of  that  kind  of  wheat  will  make 
up  the  defect,  thus : — 

Required,  the  good,  wheat  in  1000 
bushels  weighing  55  pounds  per  bushel? 


Defect,  10  Ibs.  per  bushel— 1000  in 
all. 

From  1000  bush.        55)10000(181  bush.  45  Ibs. 
Take    181  45  55 


Ans.     818  10 


450 
440 

100 
55 


45  Ibs. 

We  see  that  this  method  gives  15 
bushels  more  to  the  miller  than  the  last, 
and  28  more  than  tne  first.  It,  however, 
shows  how  much  of  the  same  kind  of 
wheat  must  be  added  to  the  1000  to 
make  1000  bushels  of  good  wheat;  viz. , 
181  bushels  45  pounds,  for  1181  bushels 
45  pounds,  weighing  55,  will  just  make 
1000  bushels  "made  up  weight."  This 
method  would  be  as  erroneous  as  calcu- 
lating discount  by  the  rule  for  interest. 

Another  method  is  to  take  two  pounds 
for  one  up  to  58,  and  pound  for  pound 
afterwards.  To  do  this  bring  the  whole 
quantity  to  pounds,  and  if  the  wheat 
weigh  57  divide  by  61;  if  56,  by  62, 
and  so  on.  This  appears  more  reason- 
able than  the  others,  as  it  makes  less 
difference  between  wheat  barely  mer- 
chantable and  that  which  is  not  quite 
so.  By  the  former  rule,  if  the  wheat 
weigh  58,  two  pounds  per  bushel  will 
make  it  up,  but  if  57,  six  pounds  are 
necessary;  by  this  rule  only  one  extra 
pound  would  be  taken.  In  subtracting 
the  odd  pounds,  the  lower  number  be- 
ing greatest  (suppose  the  wheat  to 
weigh,  say  57),  the  calculator  may  be  at 
a  loss  whether  to  take  from  57,  60  or 
63.  In  this  case,  let  the  running 
weight  be  what  it  may,  we  should  take 
from  the  weight  made  up,  as  in  example 
1,  case  2,  we  took  from  63. 

This  subject  is  of  importance  to  both 
farmers  and  millers,  and  if  they  do  not 
attend  to  it  they  deserve  to  be  cheated. 


26 


A  TABLE  FOR  MEASURING  TIMBER. 


Quarter 
Girt. 

Area. 

Quarter 
Girt. 

Area. 

Quarter 
Girt. 

Area. 

Inches. 

Feet.                Inches. 

Feet. 

Inches.                Feet. 

6 

.250              12 

1.000 

18             2.250 

6J 

.  272              12J 

1.042 

18J           2.376 

64 

.294 

12J 

1.085 

19             2.506 

6| 

317 

12f 

1.129 

194 

2.64Q 

7 

.340 

13 

1.174 

20 

2.777 

7J 

.364 

13J 

1.219 

204 

2.917 

74 

.390 

134 

1.265 

21 

3.062 

7I 

.417 

13f 

1.313 

214 

3.209 

g 

.444 

14 

1.361 

22 

3.362 

&i 

.472 

Ui 

1.410 

224 

3.516 

84 

.501 

*4 

1.460 

23 

3.673 

8| 

,531 

14| 

1.511 

234 

3.835 

Q 

.562 

15 

1.562 

24 

4.000 

yj 

.594 

16J 

1  615 

244 

4.168 

94 

.626 

1.668 

25 

4.340 

Q3 

.659 

15f 

1.722 

254 

4.516 

10 

.694 

16 

1.777 

26 

4.694 

10J 

.730 

164 

1.833 

264 

4.876 

.766 

1.890 

27 

5.062 

10| 

.£03 

16f 

1.948 

27| 

5.252 

11 

.840 

17 

2.006 

28 

5.444 

11J 

.878 

m 

2.066 

284 

5.640 

11J 

.918 

m 

2.126 

29 

5.840 

llf 

.959 

17f 

2.186 

294 

6.044 

30 

6.250 

RULE. — (BY  THE  C  ABSENTEES'  RULE.) 
Measure  the  circumference  of  the  piece 
of  timber  in  the  middle  and  take  a  quar- 
of  it  in  inches;  call  this  the  girt.  Then 
set  12  on  D  to  the  length  in  feet  on  o,  and 
against  the  girt  in  inches  on  D  you  wul 
find  the  content  in  feet  on  o. 

EXAMPLE. — If  a  piece  of  round  timber 
be  18  feet  long,  and  the  quarter  girt  24 
inches,  how  many  feet  of  timber  are 
contained  therein? 


24 -quarter  girt. 
24 

96 

48 

576  square. 
18 

4608 
576 

144)  10368  (72  feet. 
1008 

288 
288 


27 


BY  THE   TABLE. 

Against  24  stands    4.00 
Length,        18 

Product,  72.00 

Ans.  72  feet. 

By  the  Carpenters'  Rule. — 12  on  D: 
18  on  c:  24  on  D:  72  on  c. 

PROBLEM  1. — To  find  the  solid  con- 
tents of  squared  or  four-sided  timber 
by  the  Carpenters'  Rule — as  12  on  D: 
length  one:  quarter  girt  on  D:  solidity 
on  c. 

RULE  1. — Multiply  the  breadth  in  the 
middle  by  the  depth  in  the  middle,  and 
that  product  by  the  length,  for  the  solidity. 

NOTE. — If  the  tree  taper  regularly 
from  one  end  to  the  other,  half  the 
sum  of  the  breadths  of  the  two  ends 
will  be  the  breadth  in  the  middle,  and 
half  the  sum  of  the  depths  of  the  two 
ends  will  be  the  depth  in  the  middle. 

RULE  II. — Multiply  the  sum  of  the 
breadths  of  the  two  ends  by  the  sum  of  the 
depths,  to  which  add  the  product  of  the 
breadth  and  depth  of  each  end;  one  sixth 
of  this  sum  multiplied  by  the  length  will 
give  the  correct  solidity  of  any  piece  of 
squared  timber  tapering  regularly, 

PROBLEM  II. — To  find  how  much  in 
length  will  make  a  solid  foot,  or  any 
other  assigned  quantity  of  squared  tim- 
ber, of  equal  dimensions  from  end  to 
end. 

RULE. — Divide  1728,  the  solid  inches 
in  a  foot,  or  the  solidity  to  be  cut  off,  by 
the  area  of  the  end  in  inches,  and  the  quo- 
tient will  be  the  length  in  inches. 

NOTE. — To  answer  the  purpose  of  the 
above  rule,  some  carpenters'  rules  have 
a  little  table  upon  them,  in  the  follow- 
ing form,  called  a  table  of  timber  meas- 
ure. 


9     inches. 


144    36      16        9 


feet. 


side  of  the 
square. 


This  table  shows  that  if  the  side  of 
the  square  be  one  inch,  the  length  must 
be  144  feet;  if  two  inches  be  the  side 
of  the  square,  the  length  must  be  36 
feet,  to  make  a  solid  foot. 

PROBLEM  III. — To  find  the  solidity 
of  round  or  unsquared  timber. 

RULE  1. — Gird  the  timber  round  the 
middle  with  a  string;  one-fourth  part  of 
this  girt  squared  and  multiplied  by  the 
length  will  give  the  solidity. 

NOTE. — If  the  circumference  be  taken 
in  inches  and  the  length  in  feet,  divide 
the  last  product  by  144. 

RULE  II. — (BY  THE  TABLE.) — Multiply 
the  area  corresponding  to  the  quarter  girt 
in  inches,  by  the  length  of  the  piece  of 
timber  in  feet,  and  the  product  will  be  the 
solidity. 

NOTE. — If  the  quarter  girt  exceed  the 
table,  take  half  of  it,  and  four  tunes  the 
content  thus  formed  will  be  the  answer. 

How  do  you  do  when  the  timber  tapers  f 

Gird  the  timber  at  as  many  points  as 
may  be  necessary,  and  divide  the  sum 
of  the  girts  by  their  number  for  the 
mean  girt,  of  which  take  one-fifth  and 
proceed  as  before. 

If  a  tree,  girting  14  feet  at  the  thicker 
end  and  2  feet  at  the  smaller  end,  be 
24  feet  in  length,  how  many  solid  feet 
will  it  contain  ?  Ans.  122.88. 

A  tree  girts  at  five  different  places  as 
follows :  in  the  first,  9.43  feet,  in  the  sec- 
ond 7.92  feet,  in  the  third,  6.15  feet,  in 
the  fourth,  4.74  feet,  and  in  the  fifth, 
3. 16  feet;  now,  if  the  length  of  the  tree 
be  17.25  feet,  what  is  its  solidity? 

Ans.  54.42499  cubic  feet. 


or  THE 
TJNIVERSITY 


28 


OF   LOGS   FOE   SAWING. 

WJiat  is  often  necessary  for  lumber 
merchants  ? 

It  is  often  necessary  for  lumber  mer- 
chants to  ascertain  the  number  of  feet 
of  boards  which  can  be  cut  f  rom  a  given 
log,  or,  in  other  words,  to  find  how 
many  logs  will  be  necessary  to  make  a 
given  amount  of  boards, 

What  is  a  standard  board  ? 

A  standard  board  is  one  which  is  12 
inches  wide;  one  inch  thick,  and  12  feet 
long;  hence,  a  standard  board  is  one 
inch  thick  and  contains  12  square  feet. 

Wliat  is  a  standard  saw  log  ? 

A  standard  log  is  12  feet  long  and  24 
inches  in  diameter. 

How  will  you  find  the  number  of  feet 
of  boards  which  can  be  sawed  from  a 
standard  log? 

If  we  saw  off,  say  two  inches  from 
each  side,  the  log  will  be  reduced  to  a 
square  20  inches  on  a  side.  Now,  since 
a  standard  board  is  one  inch  in  thickness, 
and  since  the  saw  cuts  about  one-quar- 
ter of  an  inch  each  time  it  goes  through, 
it  follows  that  one-fourth  of  the  log 
will  be  consumed  by  the  saw.  Hence, 
we  shall  have  20xf  =  the  number  of 
boards  cut  from  the  log.  Now,  if  the 
width  of  a  board  in  inches  be  divided 
by  12,  and  the  quotient  be  multiplied 
by  the  length  in  feet,  the  product  will 
be  the  number  of  square  feet  in  the 
board.  Hence,  ||X length  of  log  in 
feet  =  the  square  feet  in  each  board. 
Therefore,  20xfX fgX length  of  log 
=  the  square  feet  in  all  the  boards  =  20 
XlOXfX&Xlength  of  log  — 20x10 
X -g- X length.  And  the  same  may  be 
shown  for  a  log  of  any  length. 

What,  then,  is  the  rule  for  finding  the 
number  of  feet  of  boards  which  can  be  cut 
from  any  log  whatever? 

From  the  diameter  of  the  log  in  inches 


subtract  four  for  the  slabs;  then  multi- 
ply the  remainder  by  half  itself,  and  the 
product  by  the  length  of  the  log  in  feet, 
and  divide  the  result  by  eight;  the  quo- 
tient will  be  the  number  of  square  feet. 
EXAMPLE  1. — What  is  the  number  of 
feet  of  boards  which  can  be  cut  from  a 
standard  log  ? 

Diameter,  24  inches. 

For  slabs,  4 

Eemainder,  20 

Half  remainder,     10 

200 
Length  of  log,        12 

8)2400 

300  =  the  number  of  feet. 

2.  How  many  feet  can  be  cut  from  a 
log  12  inches  in  diameter  and  12  feet 
long?     Ans.  48, 

3.  How  many  feet  can  be  cut  from  a 
log  20  inches  in  diameter  and  16  feet 
long?    Ans.  256, 

4.  How  many  feet  can  be-<?ut  from  a 
log  24  inches  in  diameter  and  16  feet 
long?     Ans.  400, 

5.  How  many  feet  can  be-cut  from  a 
log  28  inches  in  diameter  and  14  feet 
long?    Ans.  504, 

CAKPENTEKS'     AND    JOINERS' 
WOEK. 

In  what  does  Carpenters'  -and  Joiners' 
work  consist  ? 

Carpenters'  and  joiners'  work  is  that 
of  flooring,  roofing,  etc.,  and  is  gener- 
ally measured  by  the  square  of  100 
square  feet. 

When  is  a  roof  said  to  have  a  true 
pitch  ? 

In  carpentry  a  roof  is  said  to  have  a 
truepitch  when  the  length  of  the  rafters 
is  three-fourths  the  breadijii  of  the  build- 
ing. The  rafters  then  are  nearly  at  right 
angles.  It  is,  therefore,  customary  to 


29 


take  once  and  a  half  times  the  area  of 
the  flat  of  the  building  for  the  area  of 
the  roof. 

EXAMPLE.  1.  How  many  squares  of 
100  square  feet  each,  in  a  floor  48  feet 
6  inches  long  and  24  feet  3  inches  broad? 
Ans.  11  and  76J  sq.  ft. 

2.  A  floor  is  36  feet  6  inches  long  and 
16  feet  6  inches  broad,  how  many  squares 
does  it  contain  ?   Ans.  5  and  98J  sq.  ft. 

3.  How  many  squares  are  there  in  a 
partition  91  feet  9  inches  long  and  11  feet 
3  inches  high  ?     Ans.  10  and  32  sq.  ft- 

If  a  house  measure  within  the  walls 
52  feet  8  inches  in  length  and  30  feet 
6  inches  in  breadth,  and  the  roof  be  of 
the  true  pitch,  what  will  the  roofing 
cost  at  $1.40  per  square  ?  Ans.  $33.733. 

OF    BINS    FOB    GRAIN. 

What  is  a  bin  ? 

It  is  a  wooden  box  used  by  farmers 
for  the  storage  of  their  grain. 

Of  what  are  bins  generally  made  ? 

Their  bottoms  or  bases  are  generally 
rectangles  and  horizontal,  and  their 
sides  vertical. 

How  many  cubic  feet  are  there  in  a 
bushel  ? 

Since  a  bushel  contains  2150.4  cubic 
inches,  and  a  cubic  foot  1728  inches,  it 
follows  that  a  bushel  contains  1 J  cubic 
feet,  nearly. 

Having  any  number  of  bushels,  how 
then  will  you  find  the  corresponding 
number  of  cubic  feet. 

Increase  the  number  of  bushels  one 
fourth  itself,  and  the  result  will  be  the 
number  of  cubic  feet. 

EXAMPLE  1. — A  bin  contains  372 
bushels;  how  many  cubic  feet  does  it 
contain  ? 

372-f-4  =  93;  hence,  372x93=465 
cubic  feet. 


2.  In  a  bin   containing  400  bushels 
how  many  cubic  feet  ?    Ans.  500. 

How  will  you  find  the  number  of 
bushels  which  a  bin  of  a  given  size  will 
hold  f 

Find  the  content  of  the  bin  in  cubic 
feet;  then  diminish  the  content  by  one 
fifth,  and  the  result  will  be  the  content 
in  bushels. 

3.  A  bin  is  8  feet  long,  4  feet  wide 
and  5  feet  high;  how  many  bushels  will 
it  hold? 

8  X  4  X  5  ==  160 

then,  160  -s-  5  =  32 :     160—  32  = 
128  bushels  =  capacity  of  bin, 

4.  How  many  bushels  will  a  bin  con- 
tain which  is  7  feet  long,  3  feet  wide 
and  6  feet  in  height.     Ans.  100.8  bush. 

How  will  you  find  the  dimensions  of  a 
bin  which  shall  contain  a  given  number  of 
bushels? 

Increase  the  number  of  bushels  one 
fourth  itself,  and  the  result  will  show 
the  number  of  cubic  feet  which  the  bin 
will  contain.  Then,  when  the  two  di- 
mensions of  the  bin  are  known,  divide 
the  last  result  by  their  product,  and  the 
quotient  will  be  the  other  dimension. 

5.  what  must  be  the  height  of  a  bin 
that  will  contain  600  bushels,  its  length 
being  8  feet  and  its  breadth  4  ? 

600  --4=150;  hence,  600  +  150  = 
750=the  cubic  feet,  and  8X4=  32,  the 
product  of  the  given  dimensions.  Then, 
750  ~  32  =  23.44  feet,  the  height  of  the 
bin. 

6.  What  must  be  the  width  of  a  bin 
that   shall    contain   900    bushels,    the 
height    being    12    and  the  length   10 
feet? 

900 --4 -=225;  hence,  900x225  = 
1135  =  the  cubic  feet ;  and  12  X  10= 
120  the  product  of  the  given  dimensions. 
Then,  1155-^120=9.375  feet,  the  width 
of  the  bin. 


30 


7.  The  length  of  a  bin  is  4  feet,  its 
breadth  5  feet  6  inches,  what  must  be 
its   height    that   it    may   contain    136 
bushels  ?    Ans.  7  ft.  8  in.-f- 

8.  The   depth  of  a  bin  is  6  feet  2 
inches,  the  breadth  4  feet   8  inches; 
what  must  be  the  length  that  it  may 
contain  200  bushels?    Ans.  104  in.4- 


SLATERS'  AND    TILERS'    WORK. 

How  is  the  content  of  a  roof  found? 

In  this  work  the  content  of  the  roof 
is  found  by  multiplying  the  length  of 
the  ridge  by  the  girt  from  eave  to  eave. 
Allowances,  however,  must  be  made 
for  the  double  rows  of  slate-at  the  bot- 
tom. 

EXAMPLE  1. — The  length  of  a  slated 
roof  is  45  feet  9  inches,  and  its  girt  34 
feet  3  inches;  what  is  its  content  ?  Ans. 
1566.9375  sq.  ft. 

2.  What  will  the  tiling  of  a  barn  cost 
at  $3.40  per  square  of  100  feet,  the 
length  being  43  feet  10  inches  and 
breadth  27  feet  5  inches  on  the  flat,  the 
eave  board  projecting  16  inches  on  each 
side  and  the  roof  being  of  the  true 
pitch?  Ans.  $65.26. 

BRICKLAYERS'  WORK. 

In  how  many  ways  is  artificers'  work 
computed? 

Artificers' work  in  general  is  computed 
by  three  different  measures;  viz.  : 

1st.  The  linear  measure,  or,  as  it  is 
called  by  mechanics,  running  measure. 

2d.  Superficial  or  square  measure,  in 
which  the  computation  is  made  by  the 
square  foot,  square  yard,  or  by  the 
square  containing  100  square  feet  or 
yards. 

3d.  By  the  cubic  or  solid  measure, 
when  it  is  estimated  by  the  cubic  foot 


or  the  cubic  yard.  The  work,  however, 
is  often  estimated  in  square  measure, 
and  wie  materials  for  construction  in 
cubic  measure. 

What  proportion  do  the  dimensions  of  a 
brick  bear  to  each  other  ?  ( 

The  dimensions  of  a  brick  generally 
bears  the  following  proportions  to  each 
other,  viz. : — 

Length  —twice  the  width,  and 
Width  =twice  the   thickness;  and 
hence  the  length  is  equal  to  four  times 
the  thickness. 

What  are  the -common  dimensions  of  a 
brick  ?  How  many  cubic  inches  does  it 
contain  ? 

The  common  length  of  a  brick  is  8 
inches,  in  which  case  the  width  is  4 
inches  and  the  thickness  2  inches.  A 
brick  of  this  size  contains  8x4x2=64 
cubic  inches;  and  since  a  cubic  foot  con- 
tains 1728  cubic  inches,  we  have  1728-f- 
64=27  the  number  of  bricks  in  a  cubic 
foot. 

If  a  brick  is  9  inches  long,  what  will  be 
its  width  and  what  its  content  ? 

If  the  brick  is  9  inches  long,  then  the 
width  is  4J  inches,  and  the  thickness 
2J;  and  then  each  brick  will  contain  9 
X4JX2J=61|  cubic  inches;  and  1728 
-j-91J=19  nearly,  the  number  of  bricks 
in  a  cubic  foot.  In  the  examples  which 
follow  we  shall  suppose  the  brick  to  be 
8  inches  long. 

How  do  you  find  the  number  of  bricks 
required  to  build  a  wall  of  given  dimen- 
sions? 

1st.  Find  the  content  of  the  wall  in 
cubic  feet. 

2d.  Multiply  the  number  of  cubic 
feet  by  the  number  of  bricks,  in  a  cubic 
foot,  and  the  result  will  be  the  number 
of  bricks  required. 

EXAMPLE  1.  How  many  bricks,  of  8 
inches  in  length,  will  be  required  to 


build  a  wall  30  feet  long,  a  brick  and  a 
half  thick  and  15  feet  in  height?  Ans. 
12150. 

2.  How  many  bricks,  of    the  usual 
size,  will  be  required  to  build  a  wall  50 
feet  long,  two  bricks  thick,  and  36  feet 
in  height?    Ans.  64800. 

What  allowance  is  made  for  the  thick- 
ness of  the  mortar. 

The  thickness  of  mortar  between  the 
courses  is  nearly  a  quarter  of  an  inch, 
so  that  four  courses  will  give  nearly 
one  inch  in  height.  The  mortar,  there- 
fore, adds  nearly  one  eighth  to  the 
height;  but  as  one  eighth  is  rather  too 
large  an  allowance,  we  need  not  con- 
sider the  mortar,  which  goes  to  increase 
the  length  of  the  wall. 

3.  How  many  bricks  would   be   re- 
quired in  the  first  and  second  examples, 
if  we  make  the  proper   allowance   for 
mortar  ?    Ans.  1st.  10631J.  2d.  56700. 

How  do  bricklayers  generally  estimate 
their  work^ 

Bricklayers  generally  estimate  their 
work  at  so  much  per  thousand  bricks. 
What  is  the  cost  of  a  wall  60  feet 
long,  20  feet  high  and  2J  bricks  thick, 
at  $7.50  per  thousand — which  price  we 
suppose  to  include  the  cost  of  the 
mortar? 

If  we  suppose  the  mortar  to  occupy 
a  space  equal  to  one  eighth  the  height 
of  the  wall,  we  must  find  the  quantity 
of  bricks  under  the  supposition  that  the 
wall  was  17^  feet  in  height.  Ans.  $354. 
37J. 

In  estimating  the  bricks  for  a  house 
what  allowances  are  made? 

In  estimating  the  bricks  for  a  house, 
allowance  must  be  made  for  the  win- 
dows and  doors. 

OF    CISTERNS. 

What  are  cisterns? 

Cisterns  are  large  reservoirs  con- 
structed to  hold  water;  and,  to  be  per- 


manent, should  be  made  either  of  brick 
or  masonry.  It  frequently  occurs  that 
they  are  to  be  so  constructed  as  to  hold 
given  quantities  of  water,  and  then  it 
becomes  a  useful  and  practical  problem 
to  calculate  their  exact  dimensions. 

How  many  cubic  inches  in  a  hogshead  f 
A  hogshead  contains  63  gallons,  and 

a  gallon   contains    231    cubic   inches; 

hence,  231X63X14553,  the  number  of 

cubic  inches  in  a  hogshead. 

How  do  you  find  the  number  of  hogs- 
heads which  a  cistern  of  given  dimensions 
will  contain  ? 

1st.  Find  the  solid  content  of  the 
cistern  in  cubic  inches. 

2d.  Divide  the  content  so  found  by 
14553  and  the  quotent  will  be  the  num- 
ber of  hogsheads- 

EXAMPLE. — The  diameter  of  a  cistern 
is  6  feet  6  inches,  and  height  10  feet; 
how  many  hogsheads  does  it  contain? 

The  dimensions  reduced  to  inches  are 
78  and  120;  then,  the  content  in  cubic 
inches,  which  is  573404.832,  gives 

573404.832-4-14553r=39.40  hogsheads 
nearly. 

If  the  height  of  a  cistern  be  given,  how 
do  you  find  the  diameter,  so  that  the  cis- 
tern shall  contain  a  given  number  of  hogs- 
heads ? 

1st.  Reduce  the  height  of  the  cistern 
to  inches,  and  the  content  to  cubic 
inches. 

2d.  Multiply  the  height  by  the  deci- 
mal .7854. 

3d.  Divide  the  content  by  the  last 
result  and  extract  the  square  root  of 
the  quotient,  which  will  be  the  diameter 
of  the  cistern  in  inches. 

EXAMPLE. — The  height  of  a  cistern  is 
10  feet;  what  must  be  its  diameter  that 
it  may  contain  40  hogsheads?  Ans. 
78.6  in.  nearly. 


32 


If  the  diameter  of  a  cistern  be  given 
how  do  you  find  the  height,  so  that  the  cis- 
tern shall  contain  a  given  number  of  hogs- 
heads? 

1st.  Reduce  the  content  to  cubic 
inches. 

2d.  Reduce  the  diameter  to  inches, 
and  then  multiply  its  square  by  the 
decimal  .7854. 

3d.  Divide  the  content  by  the  last 
result,  and  the  quotient  will  be  the 
height  in  inches. 

EXAMPLE.  The  diameter  of  a  cistern 
is  8  feet;  what  must  be  its  height  that 
it  may  contain  150  hogsheads  ?  Ans.  25 
ft.  1  in.,  nearly. 


MASONS'  WORK. 

What  belongs  to  MASONRY,  and  -what 
measures  are  used? 

All  sorts  of  stone  work.  The  meas- 
ure made  use  of  is  either  superficial  or 
solid. 

Walls,  columns,  blocks  of  stone  or 
marble  are  measured  by  the  cubic  foot; 
and  pavements,  slabs,  chimney  pieces 
etc. ,  are  measured  by  the  square  or  su- 
perficial foot.  Cubic  or  solid  measure 
is  always  used  for  the  materials,  and 
the  square  measure  is  sometimes  used 
for  the  workmanship. 

EXAMPLE  1. — Required  the  solid  con- 
tent of  a  wall  53  feet  6  inches  long,  12 
feet  3  inches  high  and  2  feet  thick. 
Ans.  1310|ft. 

2.  What  is  the  solid  content  of  a 
wall  the  length  of  which  is  24  feet  3 
inches,  height  10    feet    9   inches,   and 
thickness  2  feet?     Ans.  521.375  ft. 

3.  In  a  chimney-piece  we  find  the 
following  dimensions: 


Length  of  the  mantel  and  slab  4  feet  2 

inches. 

Breadth  of  both  together,  3  ft.  2  incnes. 
Length  of  each  jam,          4  "    4     " 
Breadth  of  both,  1  "   9     " 

Required,  the  superficial  content.   Ans. 

31  ft.  10'. 


PLASTERERS3  WORK. 

How  many  kinds  of  plasierers'  work 
are  there,  and  how  are  they  measured? 

Plasterers'  work  is  of  two  kinds,  viz: 
ceiling,  which  is  plastering  on  laths; 
and  rendering,  which  is  plastering  on 
walls.  These  are  measured  separately. 

The  contents  are  estimated  either  by 
the  square  foot,  the  square  yard,  or  the 
square  of  100  feet. 

Inriched  mouldings,  etc.,  are  rated 
by  the  running  or  lineal  measure, 

In  estimating  plastering,  deductions 
are  made  for  chimneys,  doors,  windows, 
etc. 

EXAMPLE  1.  —  How  many  square 
yards  are  contained  in  a  ceiling  43  feet 
3  inches  long  and  25  feet  6  inches 
broad  ?  Ans.  122J  nearly. 

2.  What  is  the  cost  of  ceiling  a  room 
21  feel  8  inches  by  14  feet  10  inches,  at 
18  cents  per  square  yard '?   Ans.  $6.42£. 

3.  The  length  of  a  room  is  14  feet 
5  inches,  breadth  13  feet  2  inches,  and 
height  to  the  under  side  of  the  cornice 
9  feet  3  inches.     The  cornice   girts  8J 
inches,  and  projects  5  inches  from  the 
wall  on  the  upper  part   next   the  ceil- 
ing, deducting  only  for  one  door  7  feet 
by  4;  what  will  be  the  amount  of  the 
plastering? 

(  53  yds.  5  ft.  3'  6"  of  rendering. 
Ans.  1  18  yds.  5  ft.  6'  4"  of  ceiling. 

( 37  ft.  10'  9"  of  cornice. 
How  is  the  area  of  the  cornice  found  in 
the  above  examples? 

The  mean  length  of  the  cornice  both 


33 


in  the  length  and  breadth  of  tho  house 
is  found  by  taking  the  middle  line  of 
the  cornice.  Now,  since  the  cornice  pro- 
jects 5  inches  at  the  ceiling,  it  will  pro- 
ject 2J-  inches  at  the  middle  line;  and, 
therefore,  the  length  of  the  middle 
line  along  the  length  of  the  room  will 
be  14  feet,  and  across  the  room  12  feet 
9  inches.  Then  multiply  the  double  of 
each  of  these  numbers  by  the  girth,  which 
is  8J  inches,  and  the  sum  of  the  pro- 
ducts will  be  the  area  of  the  cornice. 


PAINTERS'  WORK. 

How  is  painters'  work  computed,  and 
what  allowances  are  made? 

Painters'  work  is  computed  in  square 
yards.  Every  part  is  measured  where 
the  color  lies,  and  the  measuring  line  is 
carried  into  all  the  mouldings  and  cor- 
nices. 

Windows  are  generally  done  at  so 
much  apiece.  It  is  usual  to  allow  double 
measure  for  carved  mouldings,  etc. 

EXAMPLE  1. — How  many  yards  of 
painting  in  a  room  which  is  65  feet  6 
inches  in  perimeter  and  12  feet  4  inches 
in  height  ?  Ans.  89f^  sq.  yds. 

2.  The  length  of  a  room  is  20  feet, 
its  breadth  14  feet  6  inches  and  height 
10  feet  4  inches;  how  many  yards  of 
painting  are  in  it — deducting  a  fireplace 
of  4  feet  by  4  feet  4  inches,  and  two 


windows,  each  6  feet  by  3  feet  2  inches. 
Ans.  73^V  sq.  yds, 

PAVERSrWORK. 

How  is  pavers'  work  estimated? 

Pavers'  work  is  done  by  the  square 
yard,  and  the  content  is  found  by  multi- 
plying the  length  and  breadth  together. 

EXAMPLE  1.  —  What  is  the  cost  of 
paving  a  sidewalk,  the  length  of  which 
is  35  feet  4  inches  and  breadth  8  feet  3 
inches,  at  54  cents  per  square  yard? 
Ans.  $17.48  9. 

2.  What  will  be  the  cost  of  paving 
a  rectangular  court  yard,  whose  length 
is  63  feet  and  breadth  45  feet,  at  2s.  Qd. 
per  square  yard,  there  being,  however, 
a  walk  running  lengthwise  5  feet  3 
inches  broad,  which  is  to  be  nagged 
with  stone  costing  3s  per  square  yard? 
Ans.  £40  5s. 


PLUMBERS'  WORK. 

Plumbers'  work  is  rated  at  so  much 
a  pound,  or  else  by  the  hundred  weight. 
Sheet  lead,  used  for  gutters,  etc.,  weighs 
from  6  to  12  pounds  per  square  foot. 
Leaden  pipes  vary  in  weight  according  to 
the  diameter  of  their  bore  and  thickness. 

The  following  table  shows  the  weight 
of  a  square  foot  of  sheet  lead,  accord- 
ing to  its  thickness;  and  the  common 
weight  of  a  yard  of  leaden  pipe,  ac- 
cording to  the  diameter  of  the  bore  :  — 


Thickness  of 
Lead. 

Pounds  to  a  Square 
Foot. 

Bore  of  Leaden 
Pipes. 

Pounds  per 
Yard. 

Inch. 

ft 

5.899 

Inch. 

Of 

10 

» 

6.554 

1 

12 

* 

7.373 

li 

16 

1 

8.427 

li 

18 

* 

9.831 

11 

21 

* 

11.797 

2 

24 

EXAMPLE  1. — What  weight  of  lead  of 
^  of  an  inch  in  thickness  will  cover  a 
flat  15  feet  6  inches  long,  and  10  feet  3 
inches  broad,  estimating  the  weight  at 
6  Ibs.  per  square  foot  ?  Ans.  8  cwt.  2 
qr.  1J  Ib. 

2.  What  will  be  the  cost  of  130  yards 
of  leaden  pipe  of  an  inch  and  a  half 
bore,  at  8  cents  per  pound,  supposing 
each  yard  to  weigh  18  Ibs?  Ans.  $187.20. 

3.  The  lead  used  for  a  gutter  is  12 
feet  5  inches  long  and  1  foot  3  inches 
broad,  what  is  its  weight,  supposing  it 
to  be  ^  of  an   inch  in  thickness?     Ans. 
101  Ibs.  12  oz.  13.6  dr. 

4.  What  is  the  weight  of  96  yards  of 
leaden  pipe  of  an   inch  and  a  quarter 
bore.     Ans.  13  cwt.  2  qr.  24  Ibs. 

5.  What  will  be  the  cost  of  a  sheet 
of  lead  16  feet  6  inches   long  and  10 
feet  4  inches   broad,    at   5    cents  per 
pound,  the  lead  being  &  of  an  inch  in 
thickness?    Ans.  83.81. 

WEIGHTS  AND  MEASUEES. 

TROY     WEIGHT. 

By  this  weight  gold,  silver,  platina, 
and  precious  stones  (except  diamonds) 
are  estimated. 

20  mites 1  grain. 

20  grains         ...  1  pennyw't. 

20  penny wt's 1  ounce 

12  ounces 1  pound. 

Any  quantity  of  gold  is  supposed  to 
be  divided  into  24  parts,  called  carats. 
If  pure,  it  is  said  to  be  24  carats  fine; 
if  there  is  22  parts  of  pure  gold  and  2 
parts  of  alloy  it  is  said  to  be  22  -carats 
fine.  The  standard  of  American  coin 
is  nine-tenths  pure  gold,  and  is  worth 
$20.67.  What  is  called  the  new  stan- 
dard, used  for  watch  cases,  etc.,  is  18 
carats  fine. 

The  term  carat  is  also  applied  to  a 


weight  of  3J  grains  troy,  used  in  weigh- 
ing diamonds;  it  is  divided  into  4  parts, 
called  grains;  4  grains  troy  are  thus 
equal  to  5  grains  diamond  weight. 

APOTHECARIES'   WEIGHT,    USED  IN  MEDICAL 
PRESCRIPTIONS. 

The  pound  and  ounce  of  this  weight 
are  the  same  as  the  pound  and  ounce 
troy,  but  differently  divided. 
20  grains  troy 1  scruple. 

3  scruples 1  drachm. 

8  drachms 1  ounce  troy. 

12  ounces 1  pound  troy. 

Druggists  biMj  their  goods  by 

AVOIRDUPOIS    WEIGHT. 

By  this  weight  all  goods  are  sold  ex- 
cept those  named  under  troy  weight. 

27^  grains 1  drachm, 

16  drachms 1  ounce. 

16  ounces 1  pound. 

28  pounds 1  quarter. 

4  quarters 1  hundred  weight. 

20  hundredweight.  .1  ton. 

The  grain  avoirdupois,  though  never 
used,  is  the  same  as  the  grain  in  troy 
weight.  7,000  grains  make  the  avoir- 
dupois pound,  and  5,760  grains  the 
troy  pound.  Therefore,  the  troy  pound 
is  less  than  the  avoirdupois  pound  in 
the  proportion  of  14  to  17,  nearly;  but 
the  troy  ounce  is  greater  than  the 
avoirdupois  ounce  in  the  proportion  of 
79  to  72,  nearly.  In  times  past  it  was 
the  custom  to  allow  112  pounds  for  a 
hundred  weight,  but  usage,  as  well  as 
the  laws  of  a  majority  of  the  States,  at 
the  present  time  call  100  pounds  a  hun- 
dred weight. 

APOTHECARIES'  FLUID  MEASURE. 

60  minims ....        .1  fluid  drachm. 

8  fluid  drachms.  .1  ounce  (troy). 
16  ounces  (troy).  ..1  pint. 

8  pints 1  gallon. 


35 


MEASURE  OF  CAPACITY  FOR  ALL  LIQUORS. 

5  ounces  avoirdupois,  of  water  make 
1  gill. 

4  gills 1   pint  =  34$   cubic 

inches,  nearly. 

2  pints 1  quart  =  69J   cubic 

inches. 

4  quarts 1  gallon  =  277J   do. 

inches. 

31 J  gallons 1  barrel. 

42  gallons 1  tierce. 

63  gals. ,  or  2  bbls ...  1  hogshead 

2  hogsheads 1  pipe  or  butt. 

2  pipes 1  ton. 

The  gallon  must  contain  exactly  10 
pounds  avoirdupois  of  pure  water  at  a 
temperature  of  62°,  the  barometer  being 
at  30  inches.  It  is  the  standard  unit  of 
measure  of  capacity  for  liquids  and  dry 
goods  of  every  description,  and  is  1 
larger  than  the  old  wine  measure,  ^ 
larger  than  the  old  dry  measure,  and  ^ 
less  than  the  old  ale  measure.  The  wine 
gallon  must  contain  231  cubic  inches. 

MEASURE    OF    CAPACITY  FOR   ALL   DRY  GOODS. 

4  gills 1  pint  =     34?S  cub.  in.,  nearly. 

2  pints 1  quart  =     69%  cubic  inches. 

4  quarts 1  gallon  =    277%  cubic  inches. 

2  gallons 1  peck  =    554  J<$  cubic  inches. 

4  pecks,  or  8 gals..  1  bushel  =  2150)6  cubic  inches. 

8  bushels 1  quarter  =      10%  cubic  ft.,  nearly. 

When  selling  the  following  articles  a 
barrel  weighs  as  here  stated : — 

For  rice,  600  Ibs.;  flour,  196  Ibs.; 
powder,  25  Ibs. ;  corn,  as  bought  and 
sold  in  Kentucky,  Tennessee,  etc.,  5 
bushels  of  shelled  corn;  as  bought  and 
sold  at  New  Orleans,  a  flour  barrel  full 
of  ears;  potatoes,  as  sold  in  New  York, 
a  barrel  contains  2J  bushels;  pork,  a 
barrel  is  200  Ibs.,  distinguished  in 
quality  by  "  clear,"  "  mess,"  "prime;" 
a  barrel  of  beef  is  the  same  weight. 

The  legal  bushel  of  America  is  the 
old  Winchester  measure  of  2,150.42 
cubic  inches.  The  imperial  bushel  of 
England  is  2,218.142  cubic  inches,  so 


that  32  English  bushels  are  about 
equal  to  33  of  ours. 

Although  we  are  all  the  time  talking 
about  the  price  of  grain,  etc.,  by  the 
bushel,  we  sell  by  weight,  as  follows:  — 

Wheat,  beans,  potatoes,  and  clover- 
seed,  60  Ibs.  to  the  bushel;  corn,  rye, 
flax-seed  and  onions,  56  Ibs.  ;  corn  on 
the  cob,  70  Ibs.  ;  buckwheat,  52  Ibs.  ; 
barley,  48  Ibs.  ;  hemp  seed,  44  Ibs.  ; 
timothy  seed,  45  Ibs.  ;  castor  beans,  46 
Ibs.  ;  oats,  35  Ibs.  ;  bran,  20  Ibs.  ;  blue 
grass  seed,  14  Ibs.;  salt,  the  real  weight 
of  coarse  salt  is  85  Ibs.  ;  dried  apples, 
24  Ibs.  ;  dried  peaches,  33  Ibs.  ;  accord- 
ing to  some  rules,  but  others  are  22  Ibs. 
per  bushel,  while  in  Indiana  dried 
apples  and  peaches  are  sold  by  the 
heaping  bushel;  so  are  potatoes,  turnips, 
onions,  apples,  etc.,  and  in  some  sec- 
tions oats  are  heaped.  A  bushel  of 
corn  in  the  ear  is  three  heaped  half 
bushels,  or  four  even  full. 

In  Tennessee  a  hundred  ears  of  corn 
is  sometimes  counted  as  a  bushel. 

A  hoop  18J  inches  diameter  8  inches 
deep  holds  a  Winchester  bushel.  A 
box  12  inches  square,  7  and  7^  deep, 
will  hold  half  a  bushel.  A  heaping 
bushel  is  2,815  cubic  inches. 

CLOTH    MEASURE. 

2J  inches  ........  1  nail. 

4    nails  .........  1  quarter  of  a  yard. 

4  quarters  ......  1  yard. 

FOREIGN  CLOTH  MEASURE. 

2J  quarters  ......    1  ell  Hamburg. 

3    quarters  ........  1  ell  Flemish. 

5  quarters  ........  1  ell  English. 

6  quarters  ........  1  ell  French. 

MEASURE   OF    LENGTH. 

12  inches  ......  1  foot. 

3  feet  ........  .1  yard. 

•5J  yards  ......  1  rod,  pole  or  perch. 

40  poles   .....  1  furlong. 

8  furlongs,  or  1760  yds.  1  mile. 

degree  of  a  great 


)  1 
j  ci 


miles  .....     circle  of  the  earth, 


36 


By  scientific  persons  and  revenue  offi- 
cers the  inch  is  divided  into  tenths, 
hundredth^,  etc.  Among  mechanics  the 
inch  is  divided  into  eights.  Tho  divis- 
ion of  the  inch  into  12  parts,  called 
lines,  is  not  now  ia  use. 

A  standard  English  mile,  which  is  the 
measure  that  we  use,  is  5,280  feet  in 
length,  1,760  yards,  or  320  rods.  A 
strip  one  rod  wide  and  one  mile  long  is 
two  acres;  By  this  ifc  is  easy  to  calcu- 
late the  quantity  of  land  taken  up  by 
roads,  and  also  how  much  is  wasted  by 
fences. 

GUNTER'S   CHAIN  —  USED  FOB  LAND 
MEASURE. 

7ffo  inches 1  link. 

100  links,  or  66  feet,  or  4  poles,  1  chain. 
10  chains  long  by  1  broad,  or 

10  sq.   chains 1  acre. 

Sfrchains 1  mile. 

SURFACE  MEASURE. 

144    square  inches 1  square  foot. 

9    square  feet 1  square  yard. 

30  J  square  yards  ........  1  square  rod  or 

perchc 
40    square  perches ....  1  rood. 

4    roods 1  acre. 

640    acres 1  square  mile. 

Measure  209  feet  on  each  side  and 
you  have  a  square  acre  within  an  inch. 

The  following  gives  the  comparative 
size  in  square  yards  of  acres  in  different 
countries  :— 

English  acre,  4,840  square  yards  ; 
Scotch,  6,150  ;  Irish,  7,840;  Hamburg, 
11,545  ;  Amsterdam,  9,722  ;  Dantzic, 
6,650  ;  France  (hectare),  11,960  ;  Prus- 
sia (niorgen),  3,053. 

This  difference  should  be  borne  in 
mind  in  reading  of  the  products  per 
acre  in  different  countries.  Our  land 
measure  is  that  of  England. 

GOVERNMENT  LAND  MEASURE. 

A  township — 36  sections,  each  a  mile 
square 


A  section — 640  acres. 

A  quarter  section,  half  a  mile  square 
— 160  acres. 

An  eighth  section,  half  a  mile  long, 
north  and  south,  and  a  quarter  of  a  mile 
wido — 80  acres. 

A  sixteenth  section,  a  quarter  of  a 
mile  square — 40  acres. 

The  sections  are  all  numbered  1  to 
36,  commencing  at  the  north-east  cor- 
ner thus  : — 


6 

5 

4 

3 

2 

NE 
S  W  S  E 

7 

8 

9 

10  |  11 

12 

18 

17 

16* 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

The  sections  are  divided  into  quarters, 
which  are  named  by  the  cardinal  points, 
as  in  section  1.  The  quarters  are 
divided  in  the  same  way.  The  descrip- 
tion of  a  forty-acre  lot  would  read: 
The  south  half  of  the  west  half  of  the 
south-west  quarter  of  section  1  in  town- 
ehip  24,  north  of  range  7  west,  or  as  the 
case  might  be;  and  sometimes  will  fall 
short  and  sometimes  overrun  the  num- 
ber of  acres  it  is  supposed  to  contain. 


SQUARE  MEASURE. 

FOR    CARPENTERS,    MASONS,    ETC. 

L44  square  inches 1  square  foot. 

9  sq.  ft.,  or  1,296 

sq.   in 1  square  yard. 

100  square  feet 1  sq.  of  flooring. 

roofing,  etc. 

30  J  square  yards ...    .1  square  rod. 
36  square  yards 1  rood  of  buid'g, 

"•School  section. 


37 


GEOGRAPHICAL   OR     NAUTICAL   MEASURE. 

6  feet 1  fathom. 

110  fathoms,  or  660  ft.  .1  furlong. 

6075  f  feet 1  nautical  mile. 

3  nautical  miles 1  league. 

20  leag's,  or  60  geo.  ms.  1  degree. 

360  degrees The  earth's 

circum.  =24,855  J  miles,  nearly. 
The  nautical  mile  is  795|  feet  longer 
than  the  common  mile. 

MEASURE    OF   SOLIDITY. 

1728  cubic  inches 1  cubic  foot. 

27  cubic  feet 1  cubic  yard. 

16  cubic  feet 1  cord  ft. ,    or  ft, 

of  wood. 
8  cord  ft.,   or  128 

cub.  ft 1  cord. 

40  ft.  of  round,  or  50 

ft.  of  hewn  timber  1  ton. 
42  cubic  feet 1  ton  of  shipping. 

ANGULAR     MEASURE,     OR    DIVISIONS    OF     THE 
CIRCLE. 

60  seconds 1  minute. 

60  minutes 1  degree. 

30  degrees 1  sign. 

90  degrees 1  quadrant. 

360  degrees 1  circumference. 


MEASURE    OF    TIME. 


60  seconds 1 

60  minutes 1 

24  hours 1 

7  days 1 

28  days 1 

28,  29,  30  or  31  days..  1 
12  calendar  months. .  .  1 

365  days 1 

366  days 1 

365J  days 1 

365  d.  5h.48m,  49  s.,  1 
365  d.  6h.  9m.l2s...l 


minute. 

hour. 

day. 

week. 

lunar   month. 

calen'r  month. 

year. 

com.  year. 

leap  year. 

Julian  year. 

Solar  year. 

Siderial  year. 


ROPES    AND    CABLES. 

6  feet. 1  fathom. 

120  feet 1  cable  length. 


MISCELLANEOUS     IMPORTANT     FACTS 
ABOUT  WEIGHTS  AND  MEASURES. 

BOARD    MEASURE. 

Boards  are  sold  by  superficial  measure 
at  so  much  per  foot  of  one  inch  or  less 
in  thickness,  adding  one  fourth  to  the 
price  for  each  quarter  inch  thickness 
over  an  inch. 

GRAIN  MEASURE   IJT   BULK. 

Multiply  the  width  and  length  of  the 
pile  together,  and  that  product  by  the 
height,  and  divide  by  2,150,  and  you 
have  the  contents  in  bushels. 

If  you  wish  the  contents  of  a  pile  of 
ears  of  corn,  or  roots,  in  heaped  bushels, 
ascertain  the  cubic  inches  and  divide  by 
2,818. 

CAPACITY   OF   CISTERNS    OR   WELLS. 

Tabular  view  of  the  number  of  gal- 
lons contained  in  the  clear  between  the 
brick  work  for  each  ten  inches  of  depth  : 

DIAMETER.  GALLONS. 

2  feet  equal 19 

2J  "  "  30 

3  "  "  44 

3J  "  "  60 

4  "  "  78 

4J  "  "  97 

5  "  "  : 122 

5J  "  "  148 

6  "  "  176 

6J  "  "  207 

7  "  "  240 

7J  "  "  275 

8  "  "  313 

8£  "  "  353 

9  "  "  396 

9J  "  "  461 

10  "  "  489 

11  "  "  592 

12  "  "  705 

13  "  "   827 

14  "  "  959 

15  "  " 1101 

20  "  "  1958 

25  "  "  ..3059 


38 


TO   MEASURE   COEN   IN   THE   CRIB. 

Corn  is  generally  put  up  in   crib 
made  of  rails,  but  the  rule  will  apply  to 
a  crib  of  any  size  or  kind. 

Two  cubic  feet  of  good,  sound,  drj 
corn   in  the  ear  will  make  a  bushel  o 
shelled  corn.     To  get,  then,  the  quan 
tity  of  shelled  corn  in  a  crib  of  corn  in 
the  ear,  measure  the  length,  breadth 
and  height  of  the  crib,  inside  of  the  rail 
multiply  the  length  by  the  breadth,  and 
the  product  by  the  height;  then  divide 
the     result    by   two,   and    you    have 
the  number  of  bushels  of  shelled  corn 
in  the  crib. 

In  measuring  the  height,  of  course 
the  height  of  the  corn  is  intended.  And 
there  will  be  found  to  be  a  difference  in 
measuring  corn  in  this  mode,  between 
fall  and  spring,  because  it  shrinks  very 
much  in  the  winter  and  spring,  and  set- 
tles down. 


TO  GAUGE  A  BOX  OE  GRANARY. 

RULE. — Take  the  dimensions  in  feet; 
multiply  liiem  together,  and  that  product 
by  |,  and  you  have  tJie  bushels  of  shelled 
grain.  Or, 

Place  the  dimensions  on  the  right  with 
4  under  them,  and  five  on  the  left  as  a 
divisor. 

NOTE. — This  rule  is  based  upon  the 
Winchester  bushel,  2150.4  cubic  inches. 
If  we  divide  the  number  of  cubic  inches 
in  one  foot — 1728,  by  2150,  we  have  a 
quotient  of  8,  or  ^  =  f ,  therefore,  a 
cubic  foot  holds  £  of  a  cubic  bushel; 
hence,  taking  f  of  a  cubic  feet  of  any 
space  will  give  us  bushels  of  shelled  grain. 

1.  How  many  bushels  of  wheat  will  a 
box  hold  that  is  5  feet  long,  4  feet  wide, 
and  3  feet  deep  ? 

Ans.  48  bush. 


Operation. 

5 

54 
3 
4 

48  Ans. 

2.  I  have  a  granary  10  feet  long,  8 
feet  high,  and  7  feet  wide;  how  many 
bushels  will  it  hold  ? 

Ins.  448. 

3.  If  A.  B.  Nuckolls''  granary,  12  feet 
long,  9  feet  wide,    7  feet  high,  is  just 
half  fuli,  how  many  bushels  of  wheat 
has  he  ?  Ans.  302f  bus. 

4.  A  miller  has  a  "  toll  box  "   6  feet 
long,  5   feet  wide,  3   feet  deep;    how 
much  corn  would  it  hold  ? 

Ans.  72  bus. 

5.  If  a  box  is  2  feet  in  the  clear  every 
way,  how  much  corn  would  it  hold  ? 

Ans.  6|  bus. 

6.  I  have  a  granary  12J  feet  long,  8§ 
feet  wide,    4f   feet   deep;    how   many 
bushels  will  it  hold?        Ans.  367 J  bus. 


TO  GAUGE  A  CEIB. 

RULE. — Take  the  dimensions  in  feei; 
multiply  them  together  and  that  product 
by  f ,  and  the  result  will  be  bushels.  Or, 

Place  the  dimensions  on  the  right  with 
2  under  them,  and  5  on  the  left  as  a 
divisor. 

NOTE. — Taking  f  of   the   cubic   feet 
gives  the  bushels  of  corn  in  the  ear, 
;hat   is    the   number    of    bushels   the 
'bulk"   would   ''shell  out."     Corn  is 
supposed  to  shell    out  half  the  bulk, 
lence  we  take  f  instead  of  J,  as  in  the 
preceding  rule.     The  truth  is,  there  can 
be  no  exact  rule  to  gauge  a  crib,  as  dif- 
erent  kinds  of  corn  will   ' '  shell  out " 
Lifferently,  and  a  large  crib  will  shell 
ut  more  than  a  small  one  in  proportion 


UK  THF 

((  UNIVERSITY 


39 


CALIFORHV 


to  size,  as  the  pressure  is  much  greater, 
and  will  cause  it  to  gauge  less  than  it 
will  shell  out. 

However,  the  rule  above  is  the  near- 
est correct  of  any  ever  published. 

1.  How  many  bushels  of  corn  will  a 
crib  contain  whose  dimensions  are  15 
feet  long,  ten  feet  high,  8  feet  wide  ? 

Ans.  480  bushels. 

2.  How  many   barrels   of   corn   are 
there  in  a  crib   20   feet  long,  12  feet 
high,  10  feet  wide  ?  Ans.  192  bbls. 

3.  How  many  barrels  of  corn  will  a 
crib  contain  whose  dimensions  are  20 
feet  long,  12  feet  high,  6  feet  wide  ? 

Ans.  115i  bbls. 

4.  I  have  a  crib  15|  feet  long,  1\  feet 
high,  and  6$  feet  wide;   what  are  the 
contents  in  barrels  ?          Ans.  57f  bbls. 

NOTE. — When  inches  are  given,  con- 
sider them  fractions  of  a  foot. 

5.  Stephen  Cantrell  has  a  crib  15  feet 
4  inches  long,  10  feet  6  inches  wide,  8 
feet  7  inches  high;  how  many  barrels 
will  it  hold?  Ans.  110-f-bbls. 

GAUGING  CASKS. 

RULE.  —  Take  the  distance  in  inches 
from  the  centre  of  the  bung  inside,  diag- 
onally, to  the  chine;  cube  it,  and  divide 
by  370,  and  the  quotient  will  express  the 
gallons.  Should  there  be  a  remainder, 
multiply  by  4,  and  continue  the  division 
for  quarts,  by  2,  for  pints,  etc. 

NOTE. — If  the  bung  is  not  in  the 
centre,  measure  both  ways  to  chine; 
add  the  two  results  together,  and  take 
half  the  sum;  then  proceed  as  above. 

This  standard  number  370  is  derived 
from  actual  experiment.  The  measure- 
ment of  a  regular  shaped  cask  cubed  as 
above,  divided  by  the  actual  capacity  by 
the  English  gallon  pot,  gave  the  stand- 


ard 370.  So  we  may  take  it  and  divide 
it  into  the  cube  of  any  cask,  and  we 
have  the  capacity. 

EXAMPLES. 

1.  How  many  gallons  will  a  hogshead 
hold  measuring    37    inches   from    the 
centre  of  the  bang  inside  to  the  chine  ? 

Ans.  136  gals.  3  qts.  1  pt. 

Operation. 

37  X  37  X  37  =  50653 -v- 370  =  136 

gallons. 

1st  remainder  333  X  4  =  1332-7-370 
=  3  quarts. 

2d  remainder  222  V  2  =  444-r-370= 
1  pint. 

2.  A  cask  measures  16  inches  from 
the  centre  of  the  bung,  diagonally,  to 
the  chine;  what  is  its  capacity? 

Ans.  11  gals.  2  gills. 

3.  A  cask  measures  18  inches,  diag- 
onally, to  the  chine  inside,  one  way,  and 
19  inches  the  other,  what  will  it  hold  ? 

Ans.  17  gals,  and  3-|-gills. 

4.  I  have  a  small  cask  measuring  13 
inches  to  the  chine  inside;  what  does  it 
hold  ?      Ans.  5  gals.  3  qts.  1  pt.  2  gills. 

MENTAL    OPERATIONS   IN    FRACTIONS. 

To  square  any  number  containing  J 
as  €J,  9J. 

RULE. — Multiply  the  whole  number  by 
the  next  higher  whole  number  and  annex 
J  to  the  product. 

EXAMPLE  1.  What  is  the  square  of  7J? 
Ans.  56J. 

.  We  simply  say  7  times  8  are  56,  to 
which  we  add  J. 

2.  What  will  9J  Ibs.  beef  cost  at  9J 
cts.  a  lb.? 

3.  What  will  12£  yds.  tape  cost  at  12J 
cts.  a  yd.? 

4.  What  will  5J  Ibs.  nails  cost  at5J 
cts.  a  lb.? 


40 


5.  What  will  ll^yds.  tape  cost  atllj 
cts  a  yd.  ? 

6.  What  will  19J  bu.  bran  cost  at  19J 
cts.  a  bu.  ? 

KEASON. — We  multiply  the  whole 
number  by  the  next  higher  whole  num- 
ber, because  half  of  any  number  taken 
twice  and  added  to  its  square  is  the 
same  as  to  multiply  the  given  number 
by  one  more  than  itself.  The  same  prin- 
ciple will  multiply  any  two  like  numbers 
together,  when  the  sum  of  the  fractions 
is  one,  as  8J  by  8§,  or  llf  by  llf,  etc. 
It  is  obvious  that,  to  multiply  any  num- 
ber by  any  two  fractions  whose  sum  is 
one,  the  sum  of  the  products  must  be  the 
original  number,  and  adding  the  number 
to  its  square  is  simply  to  multiply  it  by  one 
more  than  itself, — for  instance,  to  mul- 
tiply 7  J  by  7f  we  simply  say  7  times  8 
ara  56,  and  then,  to  complete  the  mul- 
tiplication, we  add,  of  course,  the  pro- 
duct of  the  fractions  (f  times  J  are  ?B), 
making  56  fa  the  answer, 

To  square  any  number  containing  \. 

RULE. — -Multiply  the  whole  number  by 
the  next  higher  whole  number  and  annex 
J  to  the  product. 

11.  What  is  the  square  of  8J?    Ans. 
72J. 

We  simply  say  9  times  8  are  72  and 
annex  J. 

12.  What  will  12J  pounds  beef  come 
to  a  12J  cents  a  pound?     Ans.    $1.56J. 

13.  What  will  6J  pounds  spike  come 
to  at  6J  cents  a  pound  ?  Ans.  42J. 

To  multiply  any  two  like  numbers  to- 
gether when  the  sum  of  the  fractions  is 
one. 

RULE. — Multiply  the  whole  number 
by  the  next  higher  whole  number,  and  to 
the  product  add  the  product  of  the  frac- 
tions. 

REMARK. — To  find  the  product  of  the 
fractions  multiply  the  numerators  to- 


gether for  a  new  numerator  and  the  de- 
nominators for  a  new  denominator. 

14.  Multiply  6J  by  6|.    Ans.  42/6. 
Explanation. — Multiply  6,  the  whole 

number,  by  7,  the  next  higher  whole 
number  =  42.  We  then  multiply  the 
numerators  of  the  fraction,  2  X  3  =  6, 
and  the  denominators  5  X  5  =  25,  mak- 
ing the  product  265,  which  we  add  to 
the  product  of  the  whole  number,  42. 

15.  Multiply  7J  by  7f .  Ans.  56f . 

16.  Multiply  llf  by  llf  Ans.  132±f 

17.  Multiply  29J  by  29f .  Ans.  870f , 
To  multiply  any  two  like  numbers  to- 
gether, each  of  which  has  a  fraction  with 
a  like  denominator,  as  3|  by  5J,  or  6f 
by  7J,  etc. 

RULE. — Add  to  the  multiplicand  the 
fraction  of  the  multiplier  and  multiply 
this  sum  by  the  whole  number;  to  the  pro- 
duct add  the  product  of  the  fractions. 

18.  Multiply  6J  by  5|,  Ans.  35ft. 

The  sum  of  6J  and  f  is  7,  so  we  sim- 
ply say  5  times  7  are  35;  to  this  we  add 
the  product  of  the  fractions,  f  times  J 
are  ^  =  35 136  Ans. 

19.  Multiply  9J  by  $|.  Ans.  78^. 
The   sum   of   9J  and  |    is  9J,  and  8 

times  9J  are  78,  to  which  add  the  pro- 
duct of  the  fractions. 

WHERE     THE      SUM      OF      THE     FRACTIONS     IS 
ONE. 

To  multiply  any  two  numbers  whose 
difference  is  one  and  the  sum  of  the 
fractions  is  one. 

RULE — Multiply  the  larger  number, 
increased  by  one,  by  the  smaller  number; 
then  square  the  fraction  of  the  larger 
number,  and  subtract  its  square  from 
one. 

PRACTICAL   EXAMPLES  FOR 
BUSINESS  MEN. 

1.  What  will  9J  Ibs.  sugar  cost  at  8| 
cts.  per  pound  ? 


Here  we  multiply  9,  increased  9J 
by  1,  by  8,  thus:  8X10  are  80,  8| 
and  set  down  the  result;  then  


from  1  we  subtract  the  square  of  80 }  J 
J  thus :  J  squared  is  ^ ,  and  1  less  y1^ 

2.  What  will  8f  bu.  coal  cost  at  7£  cts. 
abu.? 

Here  we  multiply  8,  increased  8f 

by  1,  by  7,  thus :  7  times  9  are  63,  7£ 

and   set   down  the  result;  then 

from  1  we  subtract  the  square  of  63$ 
§,  thus:  §  squared  is  f ,  and  1  less 

3.  What  will  11   ^  bu.  seed  cost  at 
$10  ^  a  bu.  ? 

Here  we  multiply  11,  increased 
byl,by  10,  thus:  10 times  12  are 

120,  and  set  down  the  result;  then  

from  1  we  substract  the  square  of  120^  f  $ 
fa,  thus:  fy  squared  is  T*^,  and  1 
less  ^  is  }|«. 

4.  How  many  square  inches  in  a  floor 
99§  inches  wide  and  98-g-  in.  long  ?  Ans_ 
9800||. 

METHOD    OF    OPERATION. 
EXAMPLE    FIRST. 

Multiply  6J  by  6J  in  a  single  line. 

Here  we  add  6  J-f- i>  which  gives  6  J 
6J;  this  multiplied  by  the  6  in  the  6J 

multiplier,    6  X  6J  gives    39,  to     

which  we  add  the  product  of  the     39.^ 
fractions;    thus   JXj   gives     *B, 
added  to  39  completes  the  pro- 
duct. 

EXAMPLE    SECOND. 

Multiply  11J  by  llf  in  a  single  line. 

Here  we  would  add  HJ-f f ,  11J 
which  gives  12 ;  this  multiplied  by  llf 

the  11  in  the  multiplier  gives  132,  

to  which  we  add  the  product  of  the  132  j36 
fractions;  thus  fXj  gives  pe,  which 
added  to  132  completes  the  product. 

EXAMPLE  THIRD. 

Multiply  12J  by  12 J  in  a  single  line. 


Here  we  add  12£-j-f  ,  which  gives  12J 
13J;  this  multiplied  by  the  12  in  12} 
the  multiplier,  12X13J,  gives  159,  - 
to  which  add  the  product  of  the  159$ 
fractions;  thus  Jxi  gives  §,  which 
added  to  159  completes  the  product. 

WHERE     THE      SUM    OF      THE      FRACTIONS     IS 
ONE. 

To  multiply  any  two  like  numbers  to- 
gether when  the  sum  of  the  fractions  is 
one, 

RULE.  —  Multiply  the  whole  number  by 
the  next  higher  whole  number,  after  which 
add  the  product  of  the  fractions. 

N.  B.  —  In  the  following  examples  the 
product  of  the  fractions  are  obtained 
first,  for  convenience  :  — 

PRACTICAL     EXAMPLES    FOR    BUSINESS     MEN. 

Multiply  3|  by  3}  in  a  single  line. 

Here  we  multiply  JXf  ,  which  3} 
gives  43ff,  and  set  down  the  result;  3J 
then  we  multiply  the  3  in  the  mul-  - 
tiplicand,  increased  by  unity,  by  12^ 
the  3  in  the  multiplier,  3X4,  which 
gives  12  and  completes  the  product. 

Multiply  7|  by  73  in  a  single  line. 
•  Here  we  multiply  f  Xf  ,  which        7$ 
gives  \6  ,  and  set  down  the  result  :         7f 
then  we  multiply  the  7  in  the  mul  -- 
tiplicand,  increased  by  unity,  by 


the  7  in  the  multiplier,  7x8,  which 
gives  56,  and  completes  the  product. 

Multiply  11J  by  llf  in  a  single  line. 

Here  we  multiply  f  X  J,  which  11  J 
gives  f,  and  set  down  the  result;  llf 
then  we  multiply  the  11  in  the  -- 
multiplicand,  increased  by  unity,  by 
132f  the  11  in  the  multiplier,  11X12, 
which  gives  132,  and  completes  the 
product. 

EXAMPLE     FOURTH. 

Multiply  16f  by  16J  in  a  single  line. 


42 


Here  we  multiply  J  X  §>  which  16  j 
gives  f ,  and  set  down  the  result;  16 J 

then  we  multiply  the  16  in  the  mul 

tiplicand,  increased  by  unity,  by  272f 
the  16  in  the  multiplier,  16X17,  which 
gives  272,  and  completes  the  product. 

EXAMPLE    FIFTH. 

Multiply  29J  by   29J  in  a  single  line. 

Here  we  multiply  JXj,  which  29  J 
gives  J,  and  set  down  the  result;  29 J 
then  we  multiply  the  29  in  the  mul- 


tiplican,  increased  by  unity,  by  870J 
the  29  in  the  multiplier,  29x30,  which 
gives  870,  and  completes  the  product. 
NOTE. — The  system  of  multiplication 
introduced  in  the  preceding  examples 
applies  to  all  numbers.  Where  the  sum 
of  the  fractions  is  one,  and  the  whole 
numbers  are  alike,  or  differ  by  one,  the 
learner  is  requested  to  study  well  these 
useful  properties  of  numbers. 

WHERE    THE    FRACTIONS    HAVE    A    LIKE    DE- 
NOMINATOR. 

To  multiply  any  two  like  numbers 
together,  each  of  which  has  a  fraction 
with  a  like  denominator,  as  4-JX4J,  or 
lliXllf,orlOfXi,etc. 

RULE.  —  Add  to  the  multiplicand  the 
fraction  of  the  multiplier,  and  multiply 
this  sum  by  the  whole  number,  after  which 
add  the  product  of  the  fractions. 

PRACTICAL     EXAMPLES     FOR     BUSINESS     MEN. 

N,  B. — In  the  following  example  the 
sum  of  the  fractions  is  one: — 

1.  "What  will  9f  Ibs.  of  beef  cost  at  9J 
cts.  a  lb.? 

The  sum  of  9|  and  J  is  10,  so  we     9f 
simply  say  9  times  10  are  90;  then     9J 
we  add  the  product  of  the  fractions,  — 
J  times  f  are  Sg.  90A36 

N.  B. — In  the  following  example  the 
sum  of  the  fractions  is  less  than  one: 


2.  What  will  8J  yds.  tape  cost  at  8f 
cts.  a  yd.? 

The  sum  of  8  J  and  |  is  8|,  so  we      8J 
simply  say  8  times  8|  are  70;  then      8J 
we  add  the  product  of  the  fractions,  - 
|  times  J  are  ^g ,  or  J.  70J 

N.  B. — In  the  following  example  the 
sum  of  the  fraction  is  greater  than 
one: — 

3.  What  will  4§  yds.  cloth  cost  at  $£ 
a  yd.? 

The  sum  of  4f  and  J  is  5  J,  so  we      4f 
simply  say  4  times  5J  are  21;  then      4£ 
we  add  the  product  of  the  fractions,  - 
£  times  f  are  gi.  21§i 

N.  B. — Where  the  fractions  have  dif- 
ferent denominators  reduce  them  to  a 
common  denominator. 

RAPID     PROCESS     FOR      MULTIPLYING-     MIXED 
NUMBEBS 

A  valuable  and  useful  rule  for  the  ac- 
countant in  the  practical  calculations  of 
the  counting  room. 

To  multiply  any  two  numbers  to- 
gether, each  of  which  involves  the  frac- 
tion \ — as  7-JX9,  etc. 

RULE. — To  the  product  of  the  whole 
numbers  add  half  their  sum,  plus  J. 

EXAMPLES  FOR  MENTAL  OPERATIONS. 

1  What  will  3  J  dozen  eggs  cost  at  7J 
cts.  a  doz.? 

Here  the  sum  of  7  and  3  is  10,      3J 
and  half  this  sum  is  5,  so  we  simply      7^ 
say  7  times  3  are  21  and  5  are  26,  - 
to  which  we  add  J.  26J 

N.  B. — If  the  sum  be  an  odd  number 
call  it  on£  less,  to  make  it  even,  and  in 
such  cases  the  fraction  must  be  -£ . 

2.  What  will  11J  Ibs.  cheese  cost  at 
9J  cts.  a  lb.?  * 


43 


3.  What  will  8J  yds.  tape  cost  at 
cts.  a  yd.? 

4.  What  will  7J  Ibs.  rice  cost  at  18  J 
cts.  a  lb.? 

5  What  will  lOJbu.  coal  cost  at  12J 
cts.  a  bu.? 

REASON. — In  explaining  the  above 
rule  we  add  half  their  sum,  because 
half  of  either  number  added  to  half  the 
other  would  be  half  their  sum,  and  we 
add  J-  because  4X4  is  J.  The  same 
principle  will  multiply  any  two  numbers 
together,  each  of  which  has  the  same 
fraction — for  instance,  if  the  fraction 
was  J  we  would  add  one-third  their  sum; 
if  £ ,  we  would  add  three-fourths  their 
sum,  etc.;  and  then,  to  complete  the 
multiplication,  we  would  add,  of  course, 
the  product  of  the  fractions. 

6.  Multiply  4g  by  4%.          Ans.  21||. 

The  sum  of  4§  and  J-  is  5J,  and  4 
«ames6Jis21;add|Xj=J|.  21f|Ans. 

To  multiply  any  two  numbers  together, 
each  of  which  involves  the  fraction  J. 

RULE. — To   the  product  of  the   whole 
numbers  add  half  their  sum,  plus% 
1.  Multiply  34X7-J.     Ans.  26J: 
Solution. — The  sum  of  3  and  7  are  10, 
and  one-half  this  sum  is  5,  so  we  say,  7 
times  3  are  21  and  5  are  26,  to  which 
we  annex  J.     26J  Ans. 

8.  What  will  7J  Ibs.  cheese  cost  at 
134  cts.  a  lb.?     Ans.  $1.01J. 

REMAKE. — If  the  sum  be  an  odd  num- 
ber call  it  one  less,  to  make  it  even;  in 
which  case  the  fraction  must  be  f . 

9.  What  will  8J  Ibs,  of  sugar  cost  at 
15J  cts.  alb.?    Ans.  $1.31  j. 

Here,  8-|-19fc=23,  being  an  odd  num- 
ber, we  make  it  one  less,  22,  one-half 
of  which  is  11.  Then  8  times  15  are 
120,  and  11  are  131,  to  which  We  add  f. 

The  same  principle  will  multiply  any 
two  numbers  together,  each  of  which 


has  the  same  fraction.  For  instance,  if 
the  fraction  was  i,  we  would  add  one- 
fifth  their  sum;  if  £,  we  would  add 
three-fourths  their  sum;  if  §,  add  two- 
thirds  their  sum,  etc.,  after  which,  of 
course,  add  the  product  of  their  frac- 
tions. 

10.  Multiply  8f  X7f.     Ans.  66|. 

The  sum  of  8  and  7  are  15,  two-thirds 
of  which  is  10.  We  then  say  8  times  7 
are  56  and  10  makes  66,  and  add  §Xf 
=*• 


INTEREST 

Is  a  sum  paid  for  the  use  of  money. 

Principal  is  a  sum  for  the  use  of 
which  interest  is  paid. 

Amount  is  the  sum  of  the  principal 
and  interest, 

Rate  per  cent.,  commonly  expressed 
decimally  as  hundredths,  is  the  sum  per 
cent,  paid  for  the  use  of  one  dollar  an- 
nually. 

Simple  Interest  is  the  sum  paid  for  the 
use  of  the  principal  only  during  the 
whole  time  of  the  loan. 

Legal  Interest  is  the  rate  per  cent,  es- 
tablished by  law- 

Usury  is  illegal  interest,  or  a  greater 
per  cent,  than  the  legal  rate. 

It  is  contended  by  many  statesmen 
that  the  rate  of  interest  should  rot  be 
established  by  statute,  but  that  money 
is  only  a  commodity  that,  like  every 
other  article  of  traffic,  should  be  gov- 
erned by  the  law  of  supply  and  demand. 
If  money  is  scarce  the  rate  would  be 
high;  if  plenty,  then  low.  But  as 
banks  and  other  great  moneyed  institu- 
tions have  the  power,  to  a  great  extent, 
of  controling  the  quantity  of  money  in 
the  market,  thereby  oppressing  the  great 
majority  of  the  people,  and  taking  ad- 
vantage of  the  times  of  scarcity,  pub- 
lic opinion,  at  least,  has  established  the 
law  of  usury. 


44 


To  find  the  interest  if  the  time  con- 
sists of  years. 

RULE. — Multiply  the  principal  by  the 
rate  per  cent.,  and  that  product  by  the 
number  of  years. 

EXAMPLE  1. — What  is  the  interest  of 
$150  for  3  years,  at  8  per  cent.? 

$150 
.08 

12.00 
3 

$36.00  Ans. 

The  decimal  for  8  per  cent,  is  .08. 
There  being  two  places  of  decimals  in 
the  multiplier  we  point  off  two  places 
in  the  product. 

To  find  the  interest  when  the  time 
consists  of  years  and  months. 

KULE.  —  Reduce  the  time  to  months. 
Multiply  the  principal  by  the  rate  per  cent. , 
divide  the  product  %  12,  and  the  quotient 
multiplied  by  the  number  of  months  will 
be  the  interest  required. 

OR  BY  CANCELLATION. — Place  the  prin- 
cipal, rate  and  time  in  months  on  the 
right  of  the  line,  and  12  on  the  left,  then 
cancel. 

2.  Find  the  interest  of  $240  for  2 
years  and  7  months,  at  7  per  cent. 

Principal,         $240 
Bate,  .07 

12)16.80 

1.40 
,     2yrs,-}-7inos.     31 


1.40 
4.20 

$43.40  Ans. 

BY  CANCELLATION. 

$240    20 


12 

20X7X" 


7 

31 
*1=$43.40.  Ans. 


BANKERS'  METHOD  OF   COMPUTING  INTEREST 
AT   6   PER    CENT.    FOR   ANY  NUMBER  OF 

DAYS. 

RULE.  — Draw  a  perpendicular  line, 
cutting  off  the  two  right-hand  figures  of 
the  $,  and  you  have  the  interest  for  60 
days  at  6  per  cent. 

NOTE. — The  figures  on  the  left  of  the 
line  are  dollars,  and  those  on  the  right 
are  decimals  of  dollars. 

EXAMPLE  1.  What  is  the  interest  of 
$423  for  60  days,  at  6  per  cent.? 

$423= the  principal. 
$4  |  23  cts.=rinterest  for  60  days. 
NOTE. — When  the  time  is  more  or  less 
than  60  days,  first  get  the  interest  for 
60  days,  and  from  that  to  the  time  re- 
quired. 

EXAMPLE  2. — What  is  the  interest  of 
$124  for  15  days  at  6  per  -cent.? 

Days.          Days. 
15=J  of  60 
$  1 24=principal . 
4)  1  |  24  cts.=interest  for  60  days. 

|  31  cts.=interest  for  15  days. 
EXAMPLE  3. — What  is  the  interest  -of 
$123.40  for  90  days,  at  6  per  cent.? 

Days.   Days.   Days. 

90  =  60      30 
$123.40=principal. 


2)1 


2340=interest  for  60  days. 
6170=interest  for  30  days. 


Ans   $  |  851=interest  for  90  days. 

EXAMPLE  4.  —  What  is  the  interest  of 
$324  for  75  days,  at  6  per  cent.? 

Daj^s.  Days.  Days. 


75  =  60       15 


$324—  principal. 


4)3 


24  cts.   interest  for  60  days. 
81  cts.  interest  for  15  days. 


Ans.  $4  |  05  cts.  interest  for  75  days. 

REMARK. — This  system  of  computing 
interest  is  very  easy  and  simple,  espec- 
ially when  the  days  are  aliquot  parts 


45 


of  60,  and  one  simple  division  will  suf- 
fice. It  is  used  extensively  by  a  large 
majority  of  our  most  prominent  bank- 
ers; and,  indeed,  is  taught  by  most  all 
commercial  colleges  as  the  shortest  sys- 
tem of  computing  interest. 

METHOD   OF  CALCULATING  AT  DIFFERENT  PER 
CENTS. 

This  principle  is  not  confined  alone 
to  6  per  cent. ,  as  many  suppose  who 
teach  and  use  it.  It  is  their  custom 
first  to  find  the  interest  at  6  per  cent. , 
and  from  that  to  other  per  cents. ;  but 
it  is  equally  applicable  for  all  per  cents. , 
from  1  to  15,  inclusive. 

The  following  table  shows  the  differ- 
ent per  cents.,  with  the  time  that  a 
given  number  of  $  will  amount  to  the 
same  number  of  cents  when  placed  at 
interest : — 

RULE. — Draw  a  perpendicular  line,  cut- 
ting off  the  two  right-hand  figures  of  $, 
and  you  have  the  interest  at  the  following 
per  cents. : — 

Interest  at  4  per  cent,  for  90  days. 

Interest  at  5  per  cent,  for  72  days. 

Interest  at  6  per  cent,  for  60  days. 

Interest  at  7  per  cent,  for  52  days. 

Interest  at  8  per  cent,  for  45  days. 

Interest  at  9  per  cent,  for  40  days. 

Interest  at  10  per  cent,  for  36  days. 

Interest  at  12  per  cent,  for  30  days. 

Interest  at  7-30  per  cent,  for  50  days. 

Interest  at  5-20  percent,  for  70 days. 

Interest  at  10-40  percent,  for 35 days. 

Interest  at  7^  per  cent,  for  48  days. 

Interest  at  4J  per  cent,  for  80  days. 

NOTE. — The  figures  on  the  left  of  the 
perpendicular  line  are  dollars,  and  on 
the  right  decimals  of  dollars.  If  the 
dollars  are  less  than  10  prefix  a  cipher. 

EXAMPLE  1. — What  is  the  interest  of 
$120  for  15  days  at  4  per  cent.? 

Days  Cays. 

$120=principal.       15=4  of  90 


6)1 


20  cts. —interest  for  90  days. 
20  cts.— interest  for  15  days. 


EXAMPLE  2.—  What  is  the  interest  of 
$132  for  13  days,  at  7  per  cent.? 

Days.  Days. 

$132—  principal.       13=J  of  52. 
4)1     32  cts.—  interest  for  52  days. 
33  cts.—  interest  for  13  days. 
EXAMPLE  3.  —  What  is  the  interest  of 
$520  for  9  days  at  8  per  cent.? 


Days. 
9— 


Days. 
of  45 


$520—  principal. 

5)5     20  cts.—  interest  for  45  days. 
$1      04  cts.—  interest  for  9  days. 
EXAMPLE  4.  —  What  is  the  interest  of 
$462  for  64  days,  at  7|  per  cent.? 

Days.  Days.  Days. 
$462=principal.        64—48+16 


3)4 

$1 


62  cts. —interest  for  48  days. 
54  cts.— interest  for  16  days. 


$6  |  16  cts.^interest  for  64  days, 
REMARK. — We  have  now  illustrated 
several  examples  by  the  different  per 
cents.,  and  if  the  student  will  study 
carefully  the  solution  to  the  above  ex- 
amples, he  will  in  a  short  time  be  very 
rapid  in  this  mode  of  computing  inter- 
est. 

NOTE. — The  preceding  mode  of  com- 
puting interest  is  derived  and  deduced 
from  the  cancelling  system,  as  the  in- 
genious student  will  readily  see.  It  is 
a  short  and  easy  way  of  finding  interest 
for  days,  when  the  days  are  even  or  ali- 
quot parts;  but  when  they  are  not 
multiples,  and  three  or  four  divisions 
are  necessary,  the  cancelling  system  is 
much  more  simple  and  easy.  We  will 
here  illustrate  an  example  to  show  the 
difference. 

Required,  the  interest  of  $420  for  49 
days,  at  6  per  cent. : — 
BANKERS'    METHOD.     CANCELLING   METHOD. 


5)1 

3) 


420—70 

6 
49 

70 

21  cts.=int.  for  3  days. 
7  cts.^int.  for  1  day.    $3.430  Ana. 


2)420  cts.=int.  for  60  days. 

6—36 
2)2 10  cts.=int.  for  30  days. 


05  cts.=int.  for  15  days. 


$3  |  43  cts.=int.  for  49  days. 


46 


The  cancelling  method  is  much  more 
brief;  we  simply  cancel  6  in  36,  and 
the  quotient  -6  into  420;  there  is  no 
devisor  left;  hence,  70X49  gives  the 
interest  at  once. 

If  the  time  had  been  15  or  20  days, 
the  bankers'  method  would  have  been 
equally  as  short,  because  15  and  20  are 
aliquot  parts  of  60.  The  superiority  of 
the  cancelling  system  above  all  others  is 
this,  it  takes  advantage  of  the  principal 
as  well  as  the  time. 

For  the  benefit  of  the  student,  and 
for  the  convenience  of  business  men,  we 
will  investigate  this  system  to  its  full 
extent,  and  explain  how  to  take  advan- 
tage of  the  principal  when  no  advantage* 
can  be  taken  of  the  days.  This  is  one 
of  the  most  important  characteristics  of 
interest,  and  very  often  saves  much 
labor.  It  should  be  used  when  the  days 
are  not  even  or  aliquot  parts. 

The  following  table  shows  the  dif- 
ferent sums  of  money  (at  the  different 
per  cents.)  that  bear  one  cent  interest  a 
day;  hence,  the  time  in  days  is  always 
the  interest  in  cents;  therefore,  to  find 
the  interest  on  any  of  the  following 
notes,  at  the  per  cent,  attached  to  it  in 
the  table,  we  have  the  following 

RULE.  —  Draw   a  perpendicular    line, 
cutting  off  the  two  right-hand  figures  of 
the  days  for  cents,  and  you  have  the  in- 
terest for  the  given  time. 
Interest  of  $90  at  4  per  cent,  for  1  day 

is  1  cent. 
Interest  of  $72  at  5  per  cent,  for  1  day 

is  1  cent. 
Interest  of  $60  at  6  per  cent,  for  1  day 

is  1  cent. 
Interest  of  $52  at  7  per  cent,  for  1  day 

is  1  cent. 
Interest  of  $45  at  8  per  cent,  for  1  day 

is  1  cent. 
Interest  of  $40  at  9  per  cent,  for  1  day 

is  1  cent. 
Interest  of  $36  at  10  per  cent,  for  1  day 

is  1  cent. 


Interest  of  $30  at  12  per  cent,  for  1  day 

is  1  cent. 
Interest  of  $50  at  7.30  per  cent,  for  1 

day  is  1  cent. 
Interest  of  $70  at  5«20  per  cent,  for  1 

day  is  1  cent. 
Interest  of  $35  at  10.40  per  cent,  for 

1  day  is  1  cent. 
Interest  of  $48  at  7J  per  cent,  for   1 

day  is  1  cent. 
Interest  of  $80  at  4J  per  cent,   for  1 

day  is  1  cent. 
Interest  of  $24  at  15  per  cent,  for  1 

day  is  1  cent. 

NOTE. — The  7-30  Government  Bonds 
are  calculated  on  the  base  of  365  days 
to  the  year,  and  the  5-20s  and  10-40s  on 
the  base  of  364  days  to  the  year. 

PROBLEMS    SOLVED    BY     BOTH    METHODS. 

We  will  now  solve  some  examples  by 
both  methods  to  further  illustrate  this 
system,  and  for  the  purpose  of  teaching 
the  pupil  how  to  use  his  judgment.  He 
will  then  have  learned  a  rule  more  val- 
uable than  all  others. 

EXAMPLE  5. — What  is  the  interest  on 
$180  for  75  days,  at  5  per  cent.  ? 

Operation  by  taking  advantage  of  the 
dollar. 

75=the  days.  $60X3  =$180. 


$0 


75  cts.=the  int.  of  $60  for  75 

days. 
3  Multiply  by  3. 


Ans.  $2  |  25cts.=the  int.  of  $180for75 
days. 

Operation  by  the  Bankers'  method. 

$180=the  principal.  60  da.  -f  15da.= 
75  da. 


4)$1 


80  cts.=the  int.  for  60  days. 
45  cts.=the  int.  for  15  days. 


Ans.  $2  |  25  cts=the  int.  for  75  days. 

By  the  first  method  we  multiplied  by 
3,  because  3X$60=$180.  By  the  sec- 
ond method  we  added  on  J,  because  60 
da.-  °da.=75  da. 


47 


N.  B. — .When  advantage  can  be  taken 
of  both  time  and  principal,  if  the  stu- 
dent wishes  to  prove  his  work  he  can 
first  work  it  by  the  Bankers5  method, 
and  then  by  taking  advantage  of  the 
principal,  or  vice  versa.  And  as  the  two 
operations  are  entirely  different,  if  the 
same  result  is  obtained  by  each,  he  may 
fairly  conclude  that  the  work  is  correct. 

PAKTIAL  PAYMENTS  ON  NOTES, 
BONDS  AND  MORTGAGES. 

To  compute  interest  on  notes,  bonds 
and  mortgages,  on  which  partial  pay- 
ments have  been  made,  two  or  three 
rules  are  given.  The  following  is  called 
the  common  rule,  and  applies  to  cases 
where  the  time  is  short,  and  payments 
made  within  a  year  of  each  other.  This 
rule  is  sanctioned  by  custom  and  com- 
mon law  ;  it  is  true  to  the  principles  of 
simple  interest,  and  requires  no  special 
enactment.  The  other  rules  are  rules 
of  law,  made  to  suit  such  cases  as  re- 
quire (either  expressed  or  implied) 
annual  interest  to  be  paid,  and,  of 
course,  apply  to  no  business  transac- 
tions closed  within  a  year. 

RULE.  —  Compute  the  interest  of  the 
principal  sum  for  the  whole  time  to  the  day 
of  settlement,  and  find  the  amount. 
Compute  the  interest  on  the  several  pay- 
ments from  the  time  each  was  paid  to  the 
day  of  settlement;  add  the  several  pay- 
ments and  the  interest  on  each  together 
and  call  the  sum  the  amount  of  the  pay- 
ments; subtracting  the  amount  of  the  pay- 
ments from  the  amount  of  the  principal 
will  leave  the  sum  due. 

EXAMPLE. — A  gave  his  note  to  B  for 
$10,000  ;  at  the  end  of  4  months  A 
paid  $6,0#0,  and  at  the  expiration  of 
another  4  months  he  paid  an  additional 
sum  of  $3,000;  how  much  did  he  owe 
B  at  the  close  of  the  year  ? 


Principal    $10,000 

Interest  for  the  whole  time . .         600 


Amount $10,600 

1st  payment $6,000. 

Interest,  8  mos. . . .     240. 

2d  payment 3,000 

Interest,  4  mos 60 


Amount $9,300 


9,300 


Pue $1,300 

PROBLEMS  IN  INTEREST. 

There  are  four  parts  or  quantities  con- 
nected with  each  operation  in  interest; 
these  are  the  Principal,  Rate  per  cent. , 
Time,  Interest  or  Amount. 

If  any  three  of  them  are  given  the 
other  may  be  found. 

Principal,  interest  and  time  given  to 
and  the  rate  per  cent. 

EXAMPLE  1. — At  what  rate  per  cent, 
must  $500  be  put  on  interest  to  gain 
$120  in  4  years  ? 

OPERATION. 
$500 

.01 


6.00 
4 

20.00)120.00(6  per  cent.    An&. 
120.00 


BY    ANALYSIS. 

The  interest  of  $1  for  the  given  time 
at  1  per  cent,  is  4  cts.  $500  will  be  500 
times  as  much=500X.04=$20.  Then 
if  $20  give  1  per  ct.,  $120  will  give  yy> 
=6  per  cent. 

RULE. — Divide  the  given  interest  by  tlie 
interest  of  the  given  sum  at  one  per  cent, 
for  the  given  time,  and  the  quotient  will 
be  the  rate  per  cent,  required- 


48 


Principal,  interest  and  rate  per  cent, 
given  to  find  the  time. 

EXAMPLE  2.— How  long  must  $500  be 
on  interest  at  6  per  cent,  to  gain 
$120? 

OPERATION. 
$500 
.06 

i-- 

30 . 00)  120.00  (4  years.     Ans. 
120.00 


ANALYSIS 

We  find  the  interest  of  $1  at  the 
given  rate  for  one  ye^r  is  six  cents. 
$500  will  be,  therefore,  500  times  as 
much  =  500 X- 06  =  $30.00.  Now,  if  it 
take  one  year  to  gain  $30,  it  will  require 
32o°  to  gain  $120=4  years.  Ans, 

EQUATION  OF  PAYMENTS. 

Equation  of  payments  is  the  process 
of  finding  the  equalized  or  average  time 
for  the  payment  of  several  sums  due  at 
different  times,  without  loss  to  either 
party. 

To  find  the  average  or  mean  time  of 
payment  when  the  several  sums  have 
the  same  date. 

KULE. — Multiply  each  payment  by  the 
time  thai  must  elapse  before  it  becomes 
due;  then  divide  the  sum  of  these  products 
by  the.  sum  of  thepayments,  and  the  quo- 
tient will  be  the  average  time  required. 

NOTE. — When  a  payment  is  to  be 
made  down  it  has  no  product,  but  it 
must  be  added  with  the  other  payments 
in  finding  the  average  time. 

EXAMPLE. — I  purchased  goods  to  the 
amount  of  $1,200;  $300  of  which  I  am 
to  pay  in  4  months,  $400  in  5  months, 
and  $500  in  8  months.  How  long  a 


credit  ought  I  to  receive  if  I  pay  the 
whole  sum  at  once?     Ans.  6  months. 

Mo.  Mo.  (       A  credit  on  $300  for  4  months 

4X300=1200  j    is  the  same  as   the  credit  on  *1 

(   for  1200  months. 
(       A  credit  on  $AOO  for  5  months 
5X400  =  2000  1   is  the  same  as  the  credit  on  $1 

(   for  2000  months. 
(       A  credit  on  $.100  foi  8  months 
8  X  500  =  4000  j   is  the  same  as  the  credit  on  $1 

(   for  4000  months. 

Therefore,  L  should  have  the 

1200)    7200 (6  nio.      same  credit  as  the  credit  on  $1 
7200  for  7200  months;  and  on  $1200, 

the  whole  sum,  one  twelve  hun- 
dredth part  of  7200  months, 
which  is  6  months. 

This  rule  is  the  one  usually  adopted 
by  merchants,  although  not  strictly 
correct;  still,  it  is  sufficiently  accurate 
for  all  practical  purposes. 

To  find  the  average  or  mean  time  of 
payment  when  the  several  sums  have 
different  dates. 

EXAMPLE.  —  Purchased  of  James 
Brown;  at  sundry  times  and  on  various 
terms  of  credit,  as  by  the  statement  an- 
nexed. When  is  the  medium  time  of 
payment? 

Jan.  1,  a  bill  amounting  to  $360,  on  3 
months'  credit. 

Jan.  15,  a  bill  amounting  to  $186,  on 

4  months'  credit. 

March  1,  a  bill  amounting  to  $450, 
on  4  months'  credit. 

May  15,  a  bill  amounting  to  $300,  on 
3  months'  credit. 

June  20,  a  bill  amounting  to  $500,  on 

5  months'  credit. 

Ans.  July  24,  or  in  115  days. 

Due,  April    1,  $360. 

"      May    15,  186  X    44  =  8184 

"     July      1,  450  X    91  =  40950 

"     Aug.   15,  300X136  =  40800 

"     Nov.   20,  500X233  =  116500 


1796-j-into)206434(114|JJ  dys 

We  first  find  the  time  when  each  of 
the  bills  will  become  due.  Then,  since 
it  will  shorten  the  operation  and  bring 
the  same  result,  we  take  the  time  when 
the  first  bill  becomes  due,  instead  of  its 
date,  for  the  period  from  which  to  com- 
pute the  average  time.  Now,  since 


49 


April  1  is  the  period  from  which  the 
average  time  is  computed,  no  time  will 
be  reckoned  on  the  first  bill,  but  the  time 
for  the  payment  of  the  second  bill  ex- 
tends 44  days  beyond  April  1,  and  we 
multiply  it  by  44. 

Proceeding  in  the  same  manner  with 
the  remaining  bills,  we  find  the  average 
time  of  payment  to  be  114  days  and  a 
fraction  from  April  1,  or  on  the  24th  of 
July. 

RULE. — Find  the  time  ivhen  each  of  the 
sums  become  due,  and  multiply  each  sum 
by  the  number  of  days  from  the  time  of 
the  earliest  payment  to  the  payment  of 
each  sum  respectively;  then  proceed  as  in 
the  last  rule,  and  the  quotient  will  be  the 
average  time  required,  in  days,  from  the 
earliest  payment. 

NOTE. — Nearly  the  same  result  may 
be  obtained  by  reckoning  the  time  in 
months. 

In  mercantile  transactions  it  is  cus- 
tomary to  give  a  credit  of  from  3  to  9 
months  on  bills  of  sale.  Merchants,  in 
settling  such  accounts  as  consist  of 
various  items  of  debit  and  credit  for 
different  times,  generally  employ  the 
following 

RULE.— Place  on  the  debtor  or  credit 
side  such  a  sum  (which  may  be  called 
MERCHANDISE  BALANCED  as  will  balance  the 
account.  Multiply  the  number  of  dollars 
in  each  entry  by  the  number  of  days 
from  the  time  the  entry  was  made  to  the 
time  of  settlement,  and  the  merchandise 
balance  by  the  number  of  days  for  which 
credit  was  given.  Then  multiply  the  dif- 
ference between  the  sum  of  the  debit  and 
the  sum  of  the  credit  products  by  the  in- 
terest o/$l  for  one  day ;  this  product  will 

be  the  INTEREST  BALANCE. 

When  the  sum  of  the  debit  products  ex- 
ceeds the  sum  of  the  credit  products  the 
interest  balance  is  in  favor  of  the  debit 
side;  but  when  the  sum  of  the  credit  pro- 


ducts exceeds  the  sum  of  the  debit  pro- 
ducts it  is  in  favor  of  the  credit  side. 
Now,  to  the  merchandise  balance  add  the 
interest  balance,  or  substract  it  as  the  case 
may  require,  and  you  obtain  the  CASH 

BALANCE. 

A  has  with  B  the  following  account. — 
1849.  Dr. 

Jan.  2.  To  merchandise,  $200 

April  20.  "  "  400 

1849.  Cr. 

Feb.  29.  By  merchandise,  $100 

May  10.  "  "  300 

If  interest  is  estimated  at  7  per  cent., 
and  a  credit  of  60  days  is  allowed  on 
the  different  sums,  what  is  the  cash  bal- 
lance  August  20,  1849?  Ans.  $206.54.' 

EXPLANATION. — Without  interest  the 
cash  balance  would  be  $200. 

The  object  of  these  changes  is  to  give 
the  learner  an  accurate  and  complete 
knowledge  of  numbers  and  of  division, 
and  the  result  is  not  the  only  object 
sought  for,  as  many  young  learners  sup- 
pose. 

How  many  times  is  75  contained  in 
575  ?  or  divide  575  by  75.  Ans.  7f . 

Divide  800  by  12J.    Quotient,  64. 

Divide  27  by  16f.  Quotient,  l^o, 
or  1ft. 

*  A  person  spent  $6  for  oranges,  at  6J 
cents  a  piece;  how  many  did  he  pur- 
chase? Ans.  96. 

When  two  or  more  numbers  are  to  be 
multiplied  together,  and  one  or  more 
of  them  have  a  cipher  on  the  right,  as 
24  by  20,  we  may  take  the  cipher  from 
one  number  and  annex  it  to  the  other 
without  affecting  the  product:  thus 
24X20  is  the  same  as  240x2;  286X 
1300=28600X13;  and  350x70x40= 
35X7X4X1000,  etc. 

Every  fact  of  this  kind,  though  ex- 
tremely simple,  will  be  very  useful  to  those 
who  wish  to  be  skillful  in  operation. 


50 


•  h  to  the    close  of   the   opera- 

nght  hand  either  of  the  multiplier  or  tion,  when  they  must  be  annexed  to  the 
multiplicand,  or  of  both,  they  may  be!  product 


TABLE  FOB  BANKING  AND  EQUATION. 

Showing  the  nuvtier  of  Days  from  any  date  in  one  Month  to  the  same  date  in  an,, 
other  Month.  Example:  How  many  days  from  the  2d  of  February  to  the 
2d  of  August  ?  Look  for  February  at  the  left  hand,  and  August  at  the  top- 
in  the  angle  is  181.  In  leap  year,  add  1  day  if  February  be  included 


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March   

334 

306 

365 
337 

28 
3fi^ 

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59 

q-i 

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151 
120 

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181 
150 

212 
181 

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304 
273 

334 

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122 

153 

184 

214 

245 

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July  

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97  A. 

oo4 

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Oocr 

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61 

92 

122 

163 

183 

August  

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184' 

A'to 
219 

<)AO 

oUtt 
970 

ooo 

Of\A 

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QQ  A 

31 

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September  

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91  9 

At  o 
9J.9 

oU4 
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664: 
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3bb 
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?>65 

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